aainstitutetext:  Dipartimento di Fisica e Astronomia, Università di Bologna e
INFN, Sezione di Bologna, via Irnerio 46, I-40126 Bologna, Italy.
bbinstitutetext:  Dipartimento di Fisica e Astronomia, Università degli Studi
di Padova e INFN, Sezione di Padova, via Marzolo 8, I-35131 Padova, Italy.
ddinstitutetext: Department of Mathematics, Kobe University, 1-1, Rokkodai,
Nada-ku, Kobe 657-8501, Japan.eeinstitutetext: Faculty of Advanced Science and
Technology, Kumamoto University, 2-39-1 Kurokami Chuo-ku Kumamoto 860-8555
Japan.

# Restrictions of Pfaffian Systems for Feynman Integrals

Vsevolod Chestnov d,e Saiei J. Matsubara-Heo b Henrik J. Munch d and Nobuki
Takayama<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

This work studies limits of Pfaffian systems, a class of first-order PDEs
appearing in the Feynman integral calculus. Such limits appear naturally in
the context of scattering amplitudes when there is a separation of scale in a
given set of kinematic variables. We model these limits, which are often
singular, via _restrictions_ of $\mathcal{D}$-modules. We thereby develop two
different restriction algorithms: one based on gauge transformations, and
another relying on the Macaulay matrix. These algorithms output Pfaffian
systems containing fewer variables and of smaller rank. We show that it is
also possible to retain logarithmic corrections in the limiting variable. The
algorithms are showcased in many examples involving Feynman integrals and
hypergeometric functions coming from GKZ systems. This work serves as a
continuation of Chestnov:2022alh .

##  1 Introduction

The evaluation of Feynman integrals in perturbative quantum field theory is a
central, albeit challenging, problem. A widely used approach is to obtain a
set of differential equations (DEQs) whose solution yields the Feynman
integrals in question. In this setting, one considers a vector $I$ of unknown
functions, dubbed the _master integrals_ , which depend on a collection of
kinematic variables $z=(z_{1},\ldots,z_{m})$, such as the masses and momenta
involved in a scattering experiment. Using the shorthand notation
$\partial_{i}=\frac{\partial}{\partial z_{i}}$, the DEQs obeyed by the master
integrals take the form

$\displaystyle\partial_{i}I=P_{i}\cdot I\quad\text{for}\quad i=1,\ldots,m\,,$
(1)

where each $P_{i}$ is an $r\times r$ matrix with entries being rational in
$z$. This is called the _Pfaffian system_ for $I$. Using (1), the unknown
vector $I$ can be analytically expressed in terms of iterated integrals
bams/1183539443 ; Duhr:2014woa , or computed by numerical integration
Liu:2022chg ; Hidding:2020ytt ; Armadillo:2022ugh for fixed values of $z$.

In practical calculations pertaining to high-energy physics, there is often a
scale separation between the $z$ variables. Namely, one variable, say $z_{1}$,
is _small_ compared to the rest (or large, in which case $1/z_{1}$ is small).
A severely incomplete list of examples includes the large top mass expansion
Davies:2021kex , the small transverse momentum expansion Alasfar:2021ppe , the
multi-Regge limit Caron-Huot:2020vlo , the soft limit in gravity Parra-
Martinez:2020dzs , various kinematic regions in effective field theories such
as the soft-collinear effective theory Becher:2014oda , and the computation of
boundary conditions using expansion by regions Dulat:2014mda in the threshold
DiVita:2014pza or massless Mastrolia:2017pfy limits. There are also
interesting limits such as threshold expansions Lee:2018ojn ; Beneke:1997zp
and limits of the $x$-parameter in the Simplified Differential Equations
approach Syrrakos:2023syr ; Papadopoulos:2014lla which involve special
kinematic configurations rather than scale-separated variables. We can model
these situations as a (typically singular!) limit $z_{1}\to 0$ on a Pfaffian
system. Starting from (1), the aim of this article is then to derive a
_simpler_ Pfaffian system which holds in the limit (see also
(haraoka2020linear, , Chapter 12) and Bytev:2022tav for similar approaches).
By simpler, we mean that the new Pfaffian system has a smaller rank than $r$,
and it depends on one variable less, because $z_{1}$ decouples from the
system.

Our algorithms are rooted in the theory of _restrictions_ (in the sense that
$z_{1}$ is being “restricted” to $0$), which is historically tied to the
theory of $\mathcal{D}$-modules. As a supplement to studying DEQs from the
viewpoint of analysis, $\mathcal{D}$-modules leverage the algebraic structure
of differential operators. When a particular $\mathcal{D}$-module is
_holonomic_ , meaning that it is associated to a system of DEQs with a finite
number of solutions, one can take advantage of the remarkable work by M.
Kashiwara kashiwara-1975 and subsequent series of works (see Hotta-Tanisaki-
Takeuchi-2008 and references therein), which generalized the theory of regular
connections by P. Deligne Deligne . This is now established as the theory of
regular holonomic systems Hotta-Tanisaki-Takeuchi-2008 . In the context of
holonomic $\mathcal{D}$-modules, we can construct bases of differential
operators, obtain relations between basis elements, count the number of
solutions to DEQs, and more. This naturally links $\mathcal{D}$-modules to
Pfaffian systems for Feynman integrals. Indeed, in this work, restrictions are
studied from two complementary points of view: one at the level of Pfaffian
systems, and the other at the level of $\mathcal{D}$-modules.

We take our inspiration from a particularly well-studied holonomic
$\mathcal{D}$-module: the GKZ hypergeometric system GKZ-1989 \- though, as we
show, the algorithms presented here also apply beyond this case. In the GKZ
framework Nasrollahpoursamami:2016 ; delaCruz:2019skx ; Klausen:2021yrt ;
Klausen:2019hrg ; Klausen:2023gui ; Tellander:2021xdz ; Pal:2021llg ;
Pal:2023kgu ; Ananthanarayan:2022ntm ; Feng:2022kgh ; Feng:2022ude ;
Feng:2019bdx ; Zhang:2023fil ; walther2022feynman ; Dlapa:2023cvx ;
Agostini:2022cgv ; Munch:2022ouq ; Klemm:2019dbm ; Bonisch:2020qmm , one
generalizes parametric representations of a Feynman integral to include extra
variables, such that now $z=(z_{1},\ldots,z_{N})$ where $N\geq m$, with the
inequality generally being strict. On one hand, this means that there are more
variables to manage. On the other hand, additional structure arises which can
be facilitated in the computation of Pfaffian systems. In particular, one can
immediately write a collection of (higher-order) differential operators which
annihilate the generalized Feynman integral. These operators generate the GKZ
$\mathcal{D}$-module. Using the fast algorithm of Hibi-Nishiyama-Takayama-2017
, it is possible to construct a basis of operators (the so-called standard
monomials) for the GKZ $\mathcal{D}$-module, which translates into a basis of
integrals for the Pfaffian system. Encoding algebraic relations between the
generators of the GKZ system into a special matrix, called the _Macaulay
matrix_ , one can readily obtain the Pfaffian matrices associated to this
basis too. This strategy for computing Pfaffian systems was carried out in the
manuscript Chestnov:2022alh , and we regard this work as continuation of the
latter.

After having made use of this GKZ technology, one seeks to dispense with the
auxiliary $z$ variables to match with the proper Feynman integral. In other
words, one asks for a _restriction_ to the case $N=m$, such that the number of
variables agree. In this work, we present two strategies with different
benefits that solve the restriction problem.

##### Pfaffian-level restriction

In this strategy, we assume that a Pfaffian system (1) is given. For instance,
the system could have been derived via integration-by-parts (IBP) following
Laporta’s algorithm Laporta:2001dd , intersection theory Mastrolia:2018uzb ;
Frellesvig:2019uqt ; Frellesvig:2020qot ; Caron-Huot:2021xqj ; Caron-
Huot:2021iev , creative telescoping vanhove-2021 , or by the Macaulay matrix
method.

Suppose that we want to restrict $z_{1}=0$, and that the Pfaffian matrices
$P_{i}$ are singular there. The first step in our protocol is to bring the
Pfaffian system into _normal form_ , meaning (roughly) that the Pfaffian
matrix $P_{1}$ has a simple pole at $z_{1}=0$, and the remaining matrices
$P_{2},\ldots$ are finite there. Normal form is achieved by Moser reduction,
an efficient algorithm involving a series of gauge transformations that we
review and give references for in Appendix B.

Next we must choose between two cases: (i) do we seek solutions for the
integrals $I$ that are holomorphic at $z_{1}=0$, or (ii) do we seek solutions
of the form $I^{(n)}(z_{2},\ldots)\log^{n}(z_{1})$ that behave logarithmically
at the origin of $z_{1}$? In case (i), one can immediately write a gauge
transformation - see (52) - which yields the restricted Pfaffian system in the
variables $z_{2},\ldots$ In case (ii), one repeats this gauge transformation
procedure several times, once for each power of $\log(z_{1})$.

This Pfaffian-level restriction protocol is efficient and applicable to any
Pfaffian system. While we have only tested it for restrictions to hyperplanes,
namely restrictions of the form $z_{1}=f(z_{2},\ldots)$ where $f$ is a linear
function of the remaining variables, we expect that it will be applicable to
restrictions onto general hypersurfaces too.

While the whole strategy ultimately only requires gauge transformations
applied to Pfaffian systems, we derive and motivate it by studying
$\mathcal{D}$-modules. The derivation is partly conjectural, but we verify it
in many examples.

#####  $\mathcal{D}$-module-level restriction

In this approach, we assume that a holonomic $\mathcal{D}$-module is given.
One example is the GKZ $\mathcal{D}$-module111Alternative
$\mathcal{D}$-modules related to Feynman integrals can also be envisaged, such
as the annihilating operators coming from conformal symmetry Henn:2023tbo and
the Yangian bootstrap, see Loebbert:2022nfu and references therein..

Traditional $\mathcal{D}$-module restriction algorithms stem from the
breakthrough work by T. Oaku Oaku-1997 . They rely on Gröbner bases in non-
commutative rings, which can become computationally expensive when many
variables are involved. In this work, instead, we extend the Macaulay matrix
method from Chestnov:2022alh to incorporate restriction of variables.

Our approach contains two steps. In the first step, we guess a basis of
differential operators for the restricted $\mathcal{D}$-module. This is done
systematically via Algorithm 4.7. In the second step, we construct a Macaulay
matrix for this basis. This matrix is polynomial in the $z$ variables, and so
we can immediately fix desired values for the $z$’s without running into
singularities. The Macaulay matrix induces a linear system of equations, the
solution to which is the restricted Pfaffian system (such a linear system can
be swiftly solved by rational reconstruction over finite fields Peraro:2019svx
; Klappert:2019emp ). The restriction method based on the Macaulay matrix is
summarized in Algorithm 4.11.

In comparison to the Pfaffian-level restriction protocol, the
$\mathcal{D}$-module level restriction does not require a Pfaffian system as
input. This is an appealing feature, because the original, un-restricted
Pfaffian system can often be computationally expensive to obtain. Furthermore,
while the Pfaffian-level restriction method is partly conjectural, the
$\mathcal{D}$-module-level restriction protocol is proven in this text to
yield the correct restriction module.

The $\mathcal{D}$-module-level protocol works for restriction to general
hypersurfaces, in addition to hyperplanes. As far as we are aware, this is the
first instance of such an algorithm. We illustrate this feature in Section 5.6
by computing the restriction of the Appell $F_{4}(x,y)$ function onto its
singular locus $xy((x-y)^{2}-2(x+y)+1)=0$.

The computer algebra software Risa/Asir was used extensively in our
$\mathcal{D}$-module computations, especially the newly developed package
mt_mm. Risa/Asir can be built from source by cloning the git repository
https://github.com/openxm-org/OpenXM. Source code for mt_mm can be found in
the directory OpenXM/src/asir-contrib/packages/src/mt_mm.

##### Symmetry relations

In addition to integration-by-parts relations, some Feynman integrals are also
linearly related to one another via _symmetry relations_. These symmetries
stem from reparametrizations of the Feynman _integrand_ resulting in the same
function after integration. Although this case is not modeled as a singular
limit on a Pfaffian system, we nevertheless observe that symmetry relations
can be treated using the Pfaffian-level restriction protocol. Our test case is
the equal-mass limit of the Pfaffian system for the three-mass two-loop
sunrise integral - see Section 5.4.

##### Outline

The remaining part of this text is structured as follows.

In Section 2, we collect preliminary material on $\mathcal{D}$-modules
associated to Euler integrals (a class of integrals which includes Feynman
integrals). This section also defines the notions of Pfaffian systems and
restrictions. To further familiarize oneself with the language of
$\mathcal{D}$-modules, we strongly recommend reading the review in Appendix A.

Section 3 introduces the central concept of a _logarithmic integrable
connection_ , a particular $\mathcal{D}$-module which represents the Pfaffian
system we seek to restrict. The story presented in this section leads us to
the three equations (52), (55) and (62) which together outline the Pfaffian-
level restriction protocol. Some steps in the construction are technical, so
in Section 3.4 we summarize how to apply the method to Feynman integrals in
practical terms. In particular, we there write the holomorphic restriction
Algorithm 3.8 and the logarithmic restriction Algorithm 3.9.

Moving on to Section 4, we begin our quest to formulate restriction algorithms
at the level of $\mathcal{D}$-modules. We begin by recalling the important
notions of weight vectors and $b$-functions in Section 4.1, as they pertain to
restriction algorithms based on Gröbner bases. This subsection culminates in
Algorithm 4.7, which systematically searches for a differential operator basis
for the restricted $\mathcal{D}$-module. Having laid out this groundwork, we
are then ready in Section 4 to formulate the $\mathcal{D}$-module-level
restriction algorithm based on the Macaulay matrix - see Algorithm 4.11. Here
we heavily rely on earlier results from the manuscript Chestnov:2022alh .

Section 5, making up the largest body of this work, contains a plethora of
examples studying restrictions on Feynman integrals and hypergeometric
functions. The goal of each example is always to derive a Pfaffian system
which holds in some limit.

We give conclusions and speculate on future developments in Section 6.

A long appendix follows the main text. We have already mentioned Appendix A
and Appendix B above. Appendix C describes improvements of algorithms
mentioned in the main text. Finally, Appendix D is a collection of proofs
which require technical details on connections and $\mathcal{D}$-modules. We
hope that this layout makes it easier to read the paper.

##  2 Preliminaries

In this section, we recall the framework of twisted cohomology groups for
Feynman integrals and fix basic notation used throughout the paper. We use
terminology on $\mathcal{D}$-modules, such as _regular holonomic_ , which is
explained in the standard textbooks Hotta-Tanisaki-Takeuchi-2008 and Borel .
See (Chestnov:2022alh, , Appendix B) for a short introduction to holonomic
$\mathcal{D}$-modules (for a pedagogical presentation which is tailored to
Feynman integrals, see also the recent article Henn:2023tbo ). Appendix A in
the present manuscript gives an introduction to tensor products of modules, as
they are used extensively throughout this text.

###  2.1 Twisted cohomology groups as $\mathcal{D}$-modules

We study Euler integrals of the form

$\displaystyle
f_{\Gamma}(z)=\int_{\Gamma}g(z;x)^{\beta_{0}}\,x_{1}^{-\beta_{1}}\cdots
x_{n}^{-\beta_{n}}\,\frac{\mathrm{d}x}{x}\quad,\quad\frac{\mathrm{d}x}{x}:=\frac{\mathrm{d}x_{1}}{x_{1}}\wedge\cdots\wedge\frac{\mathrm{d}x_{n}}{x_{n}}\,,$
(2)

which are integrated along a twisted cycle $\Gamma$, namely an integration
contour without boundary along which the branch of the integrand is specified
(aomoto2011theory, , Chapter 3). The exponents
$\beta=(\beta_{0},\ldots,\beta_{n})\in\mathbb{C}^{n+1}$ are complex
parameters. $g(z;x)$ is a Laurent polynomial in $x$ with monomial coefficients
$z_{i}$:

$\displaystyle g(z;x)=\sum_{i=1}^{N}z_{i}\,x^{\alpha_{i}}.$ (3)

This is written in multivariate exponent notation, i.e. given an integer
vector $\alpha_{i}\in\mathbb{Z}^{n}$ we set

$\displaystyle x^{\alpha_{i}}:=x_{1}^{\alpha_{i,1}}\cdots
x_{n}^{\alpha_{i,n}},$ (4)

where $\alpha_{i,j}$ stands for the $j$-th component of the vector
$\alpha_{i}$. The monomial exponents $\alpha_{i}$ induce an $(n+1)\times N$
matrix

$\displaystyle A=(a_{1}\ldots a_{N})$ (5)

with columns defined by $a_{i}:=(1,\alpha_{i})$, under the assumption that
$\texttt{Span}\\{a_{1},\ldots,a_{N}\\}=\mathbb{Z}^{n+1}$. If the variables
$z_{i}$ are all independent, the Euler integral (2) is a solution to a PDE
system called the GKZ hypergeometric system GKZ-1989 . Generalized versions of
Feynman integrals are also solutions to GKZ systems Nasrollahpoursamami:2016 ;
delaCruz:2019skx . Indeed, the Lee-Pomeransky representation Lee:2013hzt takes
the form (2), but in this case the $z_{i}$ are subject to linear constraints
determined by the topology of a Feynman diagram - some of them may be equal to
unity, or equal to each other. Therefore, we allow $z$ to vary on an affine
subspace $Y=\mathbb{A}^{m}\subset\mathbb{A}^{N}$ of dimension $m\leq N$. The
coordinates $z=(z_{1},\ldots,z_{m})$ denote an affine parametrization of $Y$.
Physically, $Y$ is the space of kinematic variables, e.g. the masses and
momenta associated to a given scattering process.

Let $\mathbb{G}_{m}$ (resp. $\mathbb{A}$) be the complex torus (resp. the
complex affine line)222 The torus $\mathbb{G}_{m}$ (resp. the complex affine
line $\mathbb{A}$) is equal to $\mathbb{C}^{\ast}:=\mathbb{C}\setminus\\{0\\}$
(resp. $\mathbb{C}$) as a set.  equipped with the Zariski topology, We define

$\displaystyle X:=\big{\\{}(z,x)\in
Y\times(\mathbb{G}_{m})^{n}\big{|}\,g(z;x)\neq 0\big{\\}}.$ (6)

$\pi:X\ \to\ Y$ is then the natural projection from the space of kinematic and
integration variables $(z,x)$ to the space of kinematic variables $z$.

Let $\mathcal{D}_{Y}$ represent the ring of linear differential operators with
polynomial coefficients in $z$. It is generated by $z_{1},\dots,z_{m}$ and
$\partial_{1},\dots,\partial_{m}$, over $\mathbb{C}$, with generators subject
to the commutation relations $[z_{i},z_{j}]=[\partial_{i},\partial_{j}]=0$,
$[\partial_{i},z_{j}]=\delta_{ij}$. Setting
$\mathcal{O}(X):=\mathbb{C}[z_{1},\dots,z_{m},x_{1}^{\pm 1},\dots,x_{n}^{\pm
1},\frac{1}{g}]$, we define an action of $\mathcal{D}_{Y}$ on
$h=h(z;x)\in\mathcal{O}(X)$ by

$\displaystyle\frac{\partial}{\partial z_{i}}\bullet h$
$\displaystyle:=\frac{\partial h}{\partial
z_{i}}+\beta_{0}\left(\frac{1}{g(z;x)}\frac{\partial g(z;x)}{\partial
z_{i}}\right)h.$ (7)

The collection of (relative) $k$-forms on $X$ is an $\mathcal{O}(X)$-module

$\displaystyle\Omega_{X/Y}^{k}:=\bigoplus_{K\subset\\{1,\ldots,n\\},\,|K|=k}{\mathcal{O}(X)}\,\mathrm{d}x^{K}\,.$
(8)

This space is acted on by the covariant derivative in integration variables
only:

$\displaystyle\nabla_{x}:=\mathrm{d}_{x}+\beta_{0}\frac{\mathrm{d}_{x}g(z;x)}{g(z;x)}\wedge-\sum_{i=1}^{n}\beta_{i}\frac{\mathrm{d}x_{i}}{x_{i}}\wedge\,.$
(9)

The $n$-th _relative de Rham cohomology group_ is then defined as

$\displaystyle M_{A}(\beta;Y)$ $\displaystyle:=\Omega_{X/Y}^{n}\ \Big{/}\
\mathrm{Im}\Bigl{(}\nabla_{x}:\,\Omega_{X/Y}^{n-1}\longrightarrow\Omega_{X/Y}^{n}\Bigr{)}\
.$ (10)

The action of $\mathcal{D}_{Y}$ on $\mathcal{O}(X)$ defined by (7) induces an
action on the quotient $M_{A}(\beta;Y)$, thereby endowing $M_{A}(\beta;Y)$
with the structure of a $\mathcal{D}_{Y}$-module. If $Y=\mathbb{A}^{N}$ then
$M_{A}(\beta;Y)$ equals the GKZ system Schulze-Walther-2009 , but we do not
assume this unless stated explicitly.

Define the _holonomic rank_ of a (left) $\mathcal{D}_{Y}$-module to be the
number of holomorphic solutions333The definition of solutions for a
$\mathcal{D}$-module is given in Appendix 441.  at a generic point in $Y$.
While this definition is _analytic_ , we also give an _algebraic_ definition
in the next subsection.

We have the following result on $M_{A}(\beta;Y)$:

######  Theorem 2.1.

$M_{A}(\beta;Y)$ is regular holonomic. If the parameter $\beta$ is generic,
its holonomic rank is given by the absolute value of the Euler characteristic
of a generic fiber, $\big{|}\chi(\pi^{-1}(z))\big{|}$. The latter equals the
number of roots of the _likelihood equation huh2013maximum _

$\displaystyle{\beta_{0}\over g(z;x)}{\partial g(z;x)\over\partial
x_{i}}-\frac{\beta_{i}}{x_{i}}=0\quad,\quad i=1,\dots,n\,,$ (11)

when $\beta$ be a generic complex vector.

See the proof in Appendix D.1.

######  Example 2.2.

Let

$\displaystyle
I_{\nu_{1},\ldots,\nu_{5}}=\int\frac{\mathrm{d}^{d}k_{1}\mathrm{d}^{d}k_{2}}{\bigl{[}k_{1}^{2}\bigr{]}^{\nu_{1}}\bigl{[}k_{2}^{2}\bigr{]}^{\nu_{2}}\bigl{[}(p_{1}-k_{1})^{2}\bigr{]}^{\nu_{3}}\bigl{[}(p_{12}-k_{12})^{2}\bigr{]}^{\nu_{4}}\bigl{[}(-p_{123}+k_{12})^{2}\bigr{]}^{\nu_{5}}}$
(12)

denote the massless two-loop N-box integral family in $d=4-2\epsilon$
dimensions with kinematics

$\displaystyle p_{i}^{2}=0\quad,\quad s=2p_{1}\cdot p_{2}\quad,\quad
t=2p_{2}\cdot p_{3}\,.$ (13)

We used the notation $p_{I}=\sum_{i\in I}p_{i}$ in the propagators.

Up to a prefactor, the Lee-Pomeransky representation of
$I_{\nu_{1},\ldots,\nu_{5}}$ can be brought into the form

$\displaystyle\int_{(0,\infty)^{5}}g(z;x)^{\epsilon-2}\,x_{1}^{\nu_{1}+\epsilon\delta}\cdots
x_{5}^{\nu_{5}+\epsilon\delta}\frac{\mathrm{d}x}{x}$ (14)

with

$\displaystyle g(z;x)$
$\displaystyle=x_{1}x_{2}\,+x_{1}x_{4}\,+x_{1}x_{5}\,+x_{2}x_{3}\,+x_{2}x_{4}\,$
$\displaystyle+x_{2}x_{5}\,+x_{3}x_{4}\,+x_{3}x_{5}\,+x_{1}x_{2}x_{4}\,+zx_{2}x_{3}x_{5}$
(15)

and $z=t/s$. Thus, the space of kinematic variables is given by

$\displaystyle Y=\\{(z_{1},\dots,z_{10})\in\mathbb{A}^{10}\mid
z_{1}=\dots=z_{9}=1\\}.$ (16)

The two parameters $\epsilon,\delta\ll 1$ are analytic regulators in the sense
of speer1969theory ; Speer1971 .

The holonomic rank of $M_{A}(\beta;Y)$ is $3$, as seen counting the number of
solutions to (11), but the holonomic rank of the GKZ system
$M_{A}(\beta;\mathbb{A}^{10})$ is $9$. $\blacksquare$

###  2.2 Zero-dimensional $\mathcal{R}$-modules and Pfaffian systems

Let $\mathcal{R}_{Y}$ denote the ring of differential operators with
coefficients in the field of _rational_ functions on $Y$ (this generalizes
$\mathcal{D}_{Y}$, which only contains polynomial coefficient functions). The
field of rational functions on $Y$ is denoted by $\mathcal{K}_{Y}$. When there
is no fear of confusion, we write $\mathcal{R}$ (resp. $\mathcal{K}$) instead
of $\mathcal{R}_{Y}$ (resp. $\mathcal{K}_{Y}$). An $\mathcal{R}$-module
naturally arises from a $\mathcal{D}_{Y}$-module ${\cal M}$ by taking a tensor
product $\mathcal{R}\otimes_{\mathcal{D}_{Y}}{\cal M}$. Note that given two
$\mathcal{D}_{Y}$-modules $\mathcal{M}_{1}$ and $\mathcal{M}_{2}$, they may
fail to be isomorphic as $\mathcal{D}_{Y}$-modules even if
$\mathcal{R}\otimes_{\mathcal{D}_{Y}}\mathcal{M}_{1}$ and
$\mathcal{R}\otimes_{\mathcal{D}_{Y}}\mathcal{M}_{2}$ are isomorphic as
$\mathcal{R}$-modules. However, as we will see, it is enough to deal with the
$\mathcal{R}$-module structure to derive a Pfaffian system.

A zero-dimensional $\mathcal{R}$-module $\mathcal{M}$ is, by definition, a
left $\mathcal{R}$-module whose dimension over $\mathcal{K}$ is finite. The
following proposition holds (dojo, , Theorem 6.9.1):

######  Proposition 2.3.

If $\mathcal{M}$ is a holonomic $\mathcal{D}_{Y}$-module, then
$\mathcal{R}\otimes_{\mathcal{D}_{Y}}\mathcal{M}$ is a zero-dimensional
$\mathcal{R}$-module.

When a $\mathcal{D}_{Y}$-module ${\cal M}$ is holonomic, its _holonomic rank_
is _algebraically_ defined to be
$\dim_{\mathcal{K}}\mathcal{R}\otimes_{\mathcal{D}_{Y}}{\cal M}$. Theorem 2.12
will show agreement between the analytic and algebraic definitions of the
holonomic rank.

Given a basis $\\{e_{1},\dots,e_{r}\\}$ of a zero-dimensional
$\mathcal{R}$-module $\mathcal{M}$, there exist _Pfaffian matrices_
$P_{i}(z)=(p_{ij}^{k})_{j,k=1}^{r}\in\mathcal{K}^{r\times r}$, $i=1,\dots,m$,
such that the identities

$\partial_{i}e_{j}=\sum_{k=1}^{r}p_{ij}^{k}e_{k}$ (17)

hold true. It follows that these matrices are subject to the integrability
condition:

$\partial_{i}\,P_{j}(z)-\partial_{j}\,P_{i}(z)=P_{i}(z)\cdot
P_{j}(z)-P_{j}(z)\cdot P_{i}(z)\>.$ (18)

Conversely, due to (17), a set of matrices $\\{P_{1}(z),\dots,P_{m}(z)\\}$
satisfying the integrability condition (18) uniquely determines a left
$\mathcal{R}$-module on $\mathcal{M}=\oplus_{i=1}^{r}\mathcal{K}e_{i}$.

For an $r$-dimensional column vector of _functions_ $I=I(z)$, the system of
PDEs

$\partial_{i}I(z)=P_{i}(z)\cdot I(z)$ (19)

is called a _Pfaffian system_. In the context of Feynman integrals, $I$ would
correspond to a basis of master integrals.

######  Example 2.4.

Let us consider the case of an ODE to give intuition on how an _algebraic_
basis $e$, consisting of differential operators with rational function
coefficients, is related to an _analytic_ basis of functions $I$. For the
moment, we set $m=1$. Take a univariate differential operator
$D=\sum_{i=0}^{r-1}a_{i}(z)\partial^{i}$, where
$\partial:=\frac{\partial}{\partial z}$ and $a_{i}\in\mathcal{K}$. $D$ then
induces an $\mathcal{R}$-module $\mathcal{M}_{D}:=\mathcal{R}/\mathcal{R}D$.
For an unknown function $f=f(z)$, the ODE

$\displaystyle D\,f=0$ (20)

is equivalent to the Pfaffian system

$\displaystyle\partial I=P\cdot I\quad,\quad
P=\left[\begin{array}[]{ccccc}0&1&\cdots&\cdots&0\\\ 0&0&1&\cdots&0\\\
\vdots&&&\ddots&\vdots\\\
-a_{0}&-a_{1}&\cdots&\cdots&-a_{r-1}\end{array}\right]\quad,\quad
I=[f,\partial f,\ldots,\partial^{r-1}f]^{T}\,.$ (25)

The corresponding algebraic basis is
$e=[1,\partial,\ldots,\partial^{r-1}]\subset\mathcal{M}_{D}$.

More generally, consider any $m>0$ and a left ideal
$\mathcal{J}\subset\mathcal{R}$ generated by differential operators
$D_{1},\dots,D_{\mu}\in\mathcal{R}$. Assume that the quotient
$\mathcal{R}$-module $\mathcal{R}/\mathcal{J}$ is zero-dimensional. Then we
can take a finite subset of monomials of partial derivatives
$\\{e_{i}=\partial^{\alpha_{i}}\\}_{i=1}^{r}$ such that its equivalence
classes in $\mathcal{R}/\mathcal{J}$ form a basis. For an unknown function
$f(z)$, the system of PDEs

$\displaystyle D_{i}\,f=0\ \ \ i=1,\dots,\mu$ (26)

is equivalent to the Pfaffian system (19) with $I=[e_{1}f,\dots,e_{r}f]^{T}$.
Let us mention that any zero-dimensional $\mathcal{R}$-module is isomorphic to
a quotient $\mathcal{R}/\mathcal{J}$ by a left ideal
$\mathcal{J}\subset\mathcal{R}$. This is proved along similar lines of
(leykin-cyclic, , Theorem 1)444The statement in this reference concerns
$\mathcal{D}$-modules, but one can write the corresponding proof for
$\mathcal{R}$-modules simply by replacing the length of a module by its
dimension over $\mathcal{K}$.. $\blacksquare$

The following example gives a free basis for the GKZ system. It is essential
for linking an operator basis to a basis of differential forms, which in turn
defines integrands for a basis of Euler integrals.

######  Example 2.5.

Let us consider the GKZ $\mathcal{D}$-module $M_{A}(\beta;\mathbb{A}^{N})$,
where we recall that $N$ equals to the number of monomials in (3). After
choosing a term order in the ring $\mathcal{R}$, a set of standard monomials
${{\tt{Std}}}$ is well-defined (dojo, , Chapter 6). Each element
$e_{i}=\partial^{\alpha_{i}}$ of ${{\tt{Std}}}$ corresponds to a cohomology
class represented by a monomial differential form

$\displaystyle\beta_{0}\,(\beta_{0}-1)\dots(\beta_{0}-|\alpha|+1)\>\frac{x^{A\,\alpha}}{g^{|\alpha|}}\>\frac{dx}{x}\>,$
(27)

where we set $A\,\alpha:=\sum_{i=1}^{N}\alpha_{i}\cdot a_{i}$. These
cohomology classes provide a free basis of $M_{A}(\beta;\mathbb{A}^{N})$ on a
non-empty open subset of $\mathbb{A}^{N}$. A Pfaffian system for $\\{e_{i}\\}$
can e.g. be obtained via a Gröbner basis computation (dojo, , Example 6.2.1).
$\blacksquare$

We note that Pfaffian matrices are basis-dependent. If we choose a different
basis $\\{\tilde{e}_{1},\dots,\tilde{e}_{r}\\}$ specified by an invertible
matrix $G=(g_{i}^{j})_{i,j=1}^{r}$ such that
${e}_{i}=\sum_{j=1}^{r}g_{i}^{j}\,\tilde{e}_{j}$, then the new Pfaffian matrix
is determined by a _gauge transformation_ :

$G[P_{i}]:=G^{-1}(P_{i}G-\partial_{i}G).$ (28)

###  2.3 Secondary-like equation

This section contains theorems which will be used in Section 5.4 to extract
certain symmetry relations among Feynman integrals.

For a left $\mathcal{R}$-module $\mathcal{M}$, we write ${\rm
End}_{\mathcal{R}}(\mathcal{M})$ for the set of $\mathcal{R}$-morphisms
$\varphi:\mathcal{M}\to\mathcal{M}$. We recall that a left
$\mathcal{R}$-module $\mathcal{M}$ is said to be simple if any of its
$\mathcal{R}$-submodules equal $\mathcal{M}$ or $0$. A proof of the following
theorem is given in Appendix D.2.

######  Theorem 2.6.

If an $\mathcal{R}$-module $\mathcal{M}$ is simple, then ${\rm
End}_{\mathcal{R}}(\mathcal{M})$ is a one-dimensional $\mathbb{C}$-vector
space.

In Section 5.4, we use the contraposition of this theorem: if
$\dim_{\mathbb{C}}{\rm End}_{\mathcal{R}}(\mathcal{M})>1$, then the
$\mathcal{R}$-module $\mathcal{M}$ has a submodule
$\mathcal{M}^{\prime}\subset\mathcal{M}$ which is neither $\mathcal{M}$ nor
$0$. It is also important to note that the converse to Theorem 2.6 is true for
semi-simple $\mathcal{R}$-modules. An $\mathcal{R}$-module is said to be semi-
simple if for any $\mathcal{R}$-submodule $\mathcal{N}$ of $\mathcal{M}$,
there exists an $\mathcal{R}$-submodule $\mathcal{N}^{\prime}$ such that
$\mathcal{M}=\mathcal{N}\oplus\mathcal{N}^{\prime}$.

######  Theorem 2.7.

When the exponent $\beta$ is generic and real, the $\mathcal{R}$-module
$\mathcal{R}\otimes_{\mathcal{D}_{Y}}M_{A}(\beta;Y)$ is a semi-simple
$\mathcal{R}$-module.

######  Theorem 2.8.

Suppose that an $\mathcal{R}$-module $\mathcal{M}$ is semi-simple. Then
$\mathcal{M}$ is simple if and only if ${\rm End}_{\mathcal{R}}(\mathcal{M})$
is a one-dimensional $\mathbb{C}$-vector space.

See the proofs of these theorems in appendices D.3, D.4 respectively.

In practice, it is useful to describe the set ${\rm
End}_{\mathcal{R}}(\mathcal{M})$ in terms of a differential equation. Fixing a
basis $\\{e_{1},\dots,e_{r}\\}$ of $\mathcal{M}$, any $\mathcal{K}$-linear map
$\varphi:\mathcal{M}\to\mathcal{M}$ is represented by an $r\times r$ matrix
$\Phi$ with entries in $\mathcal{K}$. Starting from (17) and using that
$\varphi(\partial_{i}e_{j})=\partial_{i}\varphi(e_{j})$, a short calculation
shows that $\varphi$ is $\mathcal{R}$-linear if and only if the following
differential equations hold:

$\partial_{i}\Phi=\Phi\cdot P_{i}^{T}-P_{i}^{T}\cdot\Phi\quad,\quad
i=1,\dots,m\>,$ (29)

which we call the secondary-like equation (in analogy with the secondary
equation from matsubaraheo2019algorithm ).

###  2.4 Restriction of a $\mathcal{D}$-module

Consider an affine subspace $Y^{\prime}\subset Y$, and let
$\mathcal{O}_{Y^{\prime}}$ denote the coordinate ring of $Y^{\prime}$. More
precisely, $\mathcal{O}_{Y^{\prime}}$ is given by the quotient
$\mathcal{O}_{Y}/\mathcal{I}_{Y^{\prime}}$, where $\mathcal{I}_{Y^{\prime}}$
is the ideal of $\mathcal{O}_{Y}$ which vanishes on $Y^{\prime}$. When
$Y^{\prime}=\\{z_{1}=\cdots=z_{m^{\prime}}=0\\}$, $\mathcal{I}_{Y^{\prime}}$
is generated by $z_{1},\dots,z_{m^{\prime}}$, in which case
$\mathcal{O}_{Y^{\prime}}\simeq\mathbb{C}[z_{1},\dots,z_{m}]/\mathcal{I}_{Y^{\prime}}=\mathbb{C}[z_{m^{\prime}+1},\dots,z_{m}]$.
As a note on notation, primed symbols, such as $Y^{\prime}$, will henceforth
be associated to restrictions.

For a $\mathcal{D}_{Y}$-module ${\cal M}$, we set

$\iota^{*}{\cal M}:=\mathcal{O}_{Y^{\prime}}\otimes_{\mathcal{O}_{Y}}{\cal
M}.$ (30)

We write $\iota:Y^{\prime}\hookrightarrow Y$ for the natural inclusion. By
equation (437) of the appendix, $\iota^{*}{\cal M}$ can also be written as

$\iota^{*}{\cal M}=\frac{{\cal M}}{z_{1}{\cal M}+\ldots+z_{m^{\prime}}{\cal
M}}.$ (31)

The left action of $\mathcal{D}_{Y}$ on ${\cal M}$ naturally induces that of
$\mathcal{D}_{Y^{\prime}}$ on $\iota^{*}\mathcal{M}$. We thus have

######  Definition 2.9.

The _restriction_ of a $\mathcal{D}_{Y}$-module ${\cal M}$ to $Y^{\prime}$ is
defined by the $\mathcal{D}_{Y^{\prime}}$-module $\iota^{*}\mathcal{M}$.

Recall that the $\mathcal{D}_{Y}$-module $M_{A}(\beta;Y)$ from (10)
characterizes the vector space structure of Feynman integrals. The following
proposition thus links restrictions to our study of Feynman integrals.

######  Proposition 2.10.

Let $Y^{\prime}$ be an affine subspace of $Y$. Then there is an isomorphism
$\iota^{*}M_{A}(\beta;Y)\simeq M_{A}(\beta;Y^{\prime})$ of
$\mathcal{D}_{Y^{\prime}}$-modules.

###### Proof.

It follows immediately from the definition of $M_{A}(\beta;Y)$ and the base
change formula of (Borel, , Theorem 8.4) $\Box$

######  Definition 2.11.

The ${\cal R}_{Y^{\prime}}$-_restriction_ of $\mathcal{M}$ to
$z_{1}=\ldots=z_{m^{\prime}}=0$ is defined by ${\cal
R}_{Y^{\prime}}\otimes_{{\cal D}_{Y^{\prime}}}\iota^{*}{\cal M}$. If no
confusion arises regarding the specification of $\mathcal{R}_{Y^{\prime}}$, we
simply call ${\cal R}_{Y^{\prime}}\otimes_{{\cal
D}_{Y^{\prime}}}\iota^{*}{\cal M}$ the _rational restriction_ of $\mathcal{M}$
to $z_{1}=\ldots=z_{m^{\prime}}=0$.

Pfaffian matrices for rational restrictions at the level of
$\mathcal{D}$-modules can be obtained algorithmically as follows. First,
compute a basis $e$ of the rational restriction ${\cal
R}_{Y^{\prime}}\otimes_{{\cal D}_{Y^{\prime}}}\iota^{*}{\cal M}$ as a vector
space over the rational function field $\mathcal{K}_{Y^{\prime}}$ on
$Y^{\prime}$. Second, express $\partial_{i}e$ as a linear combination of the
basis $e$. The coefficients of $e$ in this linear combination gives the
Pfaffian matrix. For more details, see Example 4.6, Theorem 4.10, Theorem 4.14
and Algorithm 4.11.

Next we relate restrictions to solution spaces of PDEs. Elements of
$\mathcal{D}_{Y}$ act on the holomorphic functions of any open set $U\subset
Y$ by $\partial_{i}\bullet f=\frac{\partial f}{\partial z_{i}}$. We denote by
${\cal O}^{an}(U)$555The superscript $an$ stands for analytic; see the
discussion in Remark A.10 for clarification. the set of the holomorphic
functions on $U$. ${\cal O}^{an}(U)$ is equipped with a left ${\cal
D}_{Y}$-module structure by this action. Let $\mathcal{I}$ be a left ideal of
$\mathcal{D}_{Y}$. It is well-known that when ${\cal M}={\cal
D}_{Y}/\mathcal{I}$ is holonomic, i.e. when $\mathcal{I}$ is a holonomic
ideal, then the space of holomorphic solutions on $U$

$\mathrm{Hom}_{{\cal D}_{Y}}({\cal M},{\cal O}^{an}(U))=\left\\{f\in{\cal
O}^{an}(U)\bigm{|}\ell\bullet f=0\mbox{ for all
$\ell\in\mathcal{I}$}\right\\}$ (32)

is a finite-dimensional vector space over $\mathbb{C}$ (see Appendix A). Let
$c$ be a point in $Y$. We denote by $B(c,R)$ the open ball with the center $c$
and radius $R$. There exists a number $R_{0}>0$ such that
$\mathrm{dim}_{\mathbb{C}}\,\mathrm{Hom}_{\mathcal{D}_{Y}}({\cal M},{\cal
O}^{an}(B(c,R)))$ is a constant for any $R\leq R_{0}$. We call it the space of
the holomorphic solutions of ${\cal M}$ at $z=c$. The following result is
fundamental.

######  Theorem 2.12.

See the expositions (dojo, , Chapter 6), coutinho , (SST, , Section 5.2).

  1. 1.

The left ${\cal D}_{Y^{\prime}}$-module $\iota^{*}{\cal M}$ is a holonomic
$\mathcal{D}_{Y^{\prime}}$-module.

  2. 2.

Let ${\cal M}$ be regular holonomic and let $c\in Y^{\prime}$ be a point
outside of the singular locus of $\iota^{*}{\cal M}$. Then the holonomic rank
of $\iota^{*}{\cal M}$ agrees with the dimension of the space of holomorphic
solutions $\mathrm{dim}_{\mathbb{C}}\,\mathrm{Hom}_{\mathcal{D}_{Y}}({\cal
M},{\cal O}^{an}(B(c,R_{0})))$ when $R_{0}>0$ is sufficiently small. In
particular, if ${\cal M}$ takes the form ${\cal
M}=\mathcal{D}_{Y}/\mathcal{I}$ for some left ideal
$\mathcal{I}\subset\mathcal{D}_{Y}$, we have

$\mathrm{dim}_{\mathbb{C}}\,\mathrm{Hom}_{\mathcal{D}_{Y}}\Bigl{(}{\cal
M},{\cal
O}^{an}\bigl{(}B(c,R_{0})\bigr{)}\Bigr{)}=\mathrm{dim}_{\mathbb{C}}\,\frac{\mathcal{D}_{Y}}{\mathcal{I}+(z_{1}-c_{1})\mathcal{D}_{Y}+\ldots+(z_{m}-c_{m})\mathcal{D}_{Y}}$
(33)

by setting $m^{\prime}=m$. In other words, we set $Y^{\prime}=\\{c\\}$.

  3. 3.

If $c\in Y$ does not belong to the singular locus of ${\cal M}$,
$\mathrm{dim}_{\mathbb{C}}\,\mathrm{Hom}_{\mathcal{D}_{Y}}({\cal M},{\cal
O}^{an}(B(c,R_{0})))$ is equal to the holonomic rank of ${\cal M}$.

Let ${\cal D}_{Y^{\prime}}^{k}:=\overbrace{{\cal
D}_{Y^{\prime}}\times\cdots\times{\cal D}_{Y^{\prime}}}^{k}$ be a free
$\mathcal{D}_{Y^{\prime}}$-module. It is well-known that the restriction
$\iota^{*}{\cal M}$ can be expressed as $\iota^{*}{\cal
M}=\mathcal{D}^{k}_{Y^{\prime}}/\mathcal{J}$ for some $k$, where $\mathcal{J}$
is a left ${\cal D}_{Y^{\prime}}$-submodule of $\mathcal{D}_{Y^{\prime}}^{k}$
dojo ; coutinho . The restriction can also be expressed in terms of a Göbner
basis as in (96). The holonomic rank of $\iota^{*}{\cal M}$ can be determined
by computing a Gröbner basis of ${\cal R}_{Y^{\prime}}\mathcal{J}$ in a free
left ${\cal R}_{Y^{\prime}}$-module ${\cal R}_{Y^{\prime}}^{k}$666For
introductions to Gröbner bases of a submodule in a free module, see (dojo, ,
Section 3.5), (adams, , Chapter 3), and Appendix A. The first two references
discuss modules over a polynomial ring, but it is easy to generalize their
exposition to left modules over ${\cal D}_{Y^{\prime}}$ or ${\cal
R}_{Y^{\prime}}$..

For a given holonomic ${\cal D}_{Y^{\prime}}$-module ${\cal
D}_{Y^{\prime}}^{k}/\mathcal{J}$, there exists a left ideal $\mathcal{I}$ of
${\cal D}_{Y^{\prime}}$ such that ${\cal D}_{Y^{\prime}}/\mathcal{I}$ is
isomorphic to ${\cal D}_{Y^{\prime}}^{k}/\mathcal{J}$ as a ${\cal
D}_{Y^{\prime}}$-module. An algorithmic construction of $\mathcal{I}$, relying
on the so-called _cyclic vector_ , is given in leykin-cyclic . Our Remark A.9
gives an easy, albeit inexact, derivation. The cyclic vector also allows for
the construction of an ideal associated to a Pfaffian system.

##  3 Restriction of a Pfaffian system

In this section, we present our algorithm for the restriction of a generic
Pfaffian system. In Section 3.1, we revise key points in the theory of
integrable connections. Then in Section 3.2 and Section 3.3, we show the
restriction procedure for a logarithmic connection and the corresponding
solution of the Pfaffian system. Finally, in Section 3.4, we apply the
developed theory to the Pfaffian systems coming from Feynman integrals.

###  3.1 Integrable connection

The arguments we present in this section are Zariski local, meaning that we
always work on a Zariski open subset $Y_{0}$ of the total space $Y$. Let
$\mathcal{O}_{Y_{0}}$ denote the ring of regular functions on $Y_{0}$. When
$Y_{0}$ is a complement of a hypersurface $\\{f=0\\}$ defined by
$f\in\mathcal{O}_{Y}\setminus\\{0\\}$, $\mathcal{O}_{Y_{0}}$ is given by
$\mathbb{C}[z_{1},\dots,z_{m},\frac{1}{f}]$. We set
$\mathcal{D}_{Y_{0}}:=\mathcal{O}_{Y_{0}}\otimes_{\mathcal{O}_{Y}}\mathcal{D}_{Y}$.
The ring structure of $\mathcal{D}_{Y}$ induces that of $\mathcal{D}_{Y_{0}}$.
An _integrable connection_ ${\cal M}$ on $Y_{0}$ is a free
$\mathcal{D}_{Y_{0}}$-module of finite rank as an
$\mathcal{O}_{Y_{0}}$-module777 In the standard literature, an integrable
connection is usually defined as a $\mathcal{D}_{Y}$-module which is locally
free as an $\mathcal{O}_{Y}$-module. However, a locally free
$\mathcal{O}_{Y}$-module is a free module when it is restricted to an open
subset. As we are only interested in the structure of $\mathcal{R}$-modules in
this paper, the global structure of a locally free module does not play any
role. Hence, our definition of an integrable connection does not involve
locally free modules. . More concretely, ${\cal M}$ is isomorphic to
$\mathcal{O}_{Y_{0}}^{r}$ as an $\mathcal{O}_{Y_{0}}$-module for some integer
$r$. The action of $\partial_{i}$ (for $i=1,\ldots,m$) is determined by an
$r\times r$ matrix $P_{i}(z)$ with entries in $\mathcal{O}_{Y_{0}}$ according
to the formula (17), where $\\{e_{1},\dots,e_{r}\\}$ is any
$\mathcal{O}_{Y_{0}}$-free basis. The integrability condition (18) follows
straightforwardly for the matrices $P_{i}(z)$ obtained in this way.
Conversely, any set of matrices $\\{P_{i}\\}_{i=1}^{r}$ with entries in
$\mathcal{O}_{Y_{0}}$ satisfying the integrability condition (18) gives rise
to a connection

$\displaystyle\nabla_{i}=\partial_{i}-P_{i}$ (34)

by the formula (17). In this representation, the integrability condition
translates into the commutator relation $[\nabla_{i},\nabla_{j}]=0\>$.

Any holonomic $\mathcal{D}_{Y}$-module is an integrable connection on a non-
empty open subset $Y_{0}\>$. For a $\mathcal{D}_{Y}$-module ${\cal M}\>$, we
define a left $\mathcal{D}_{Y_{0}}$-module ${\cal M}|_{Y_{0}}$ by ${\cal
M}|_{Y_{0}}:=\mathcal{O}_{Y_{0}}\otimes_{\mathcal{O}_{Y}}{\cal M}$. As the
following proposition shows, studying the $\mathcal{R}$-module structure of a
$\mathcal{D}_{Y}$-module is the same as restricting it onto an open subset
wherein it is an integrable connection.

######  Proposition 3.1.

Let ${\cal M}_{1},{\cal M}_{2}$ be a pair of $\mathcal{D}_{Y}$-modules. ${\cal
M}_{1}$ and ${\cal M}_{2}$ are isomorphic as $\mathcal{R}$-modules if and only
if there is a non-empty open subset $Y_{0}\subset Y$ such that ${\cal
M}_{1}|_{Y_{0}}$ and ${\cal M}_{2}|_{Y_{0}}$ are both integrable connections
and isomorphic as $\mathcal{D}_{Y_{0}}$-modules.

Next, we recall the definition of a logarithmic connection. To simplify the
exposition, let $Y^{\prime}$ be an affine hyperplane of $Y$. By an affine
linear change of coordinates, we may assume that $Y^{\prime}=\\{z_{1}=0\\}\>$.
We write $\mathcal{D}_{Y}(\log Y^{\prime})$ for the subring of
$\mathcal{D}_{Y}$ generated as

$\displaystyle\mathcal{D}_{Y}(\log Y^{\prime}):=\mathbb{C}\langle
z_{1},\dots,z_{m},\,z_{1}\partial_{1}\,,\partial_{2},\dots,\partial_{m}\rangle\>.$
(35)

The use of the symbol $\log Y^{\prime}$ is justified because (37) below has a
logarithmic singularity along $Y^{\prime}$. We further introduce
$Y_{0}^{\prime}:=Y_{0}\cap Y^{\prime}$ and assume
$Y_{0}^{\prime}\neq\varnothing\>$. The relations between the sets $Y\>$,
$Y^{\prime}\>$, $Y_{0}\>$, and $Y_{0}^{\prime}$ are shown in Figure 1.

Figure 1: The different spaces $Y,Y^{\prime},Y_{0}$ and $Y_{0}^{\prime}$
wherein the $z$-variables live. The coordinates $\\{z_{1},\ldots,z_{m}\\}$ and
$\\{z_{m^{\prime}+1},\ldots,z_{m}\\}$ parameterize $Y$ and $Y^{\prime}$
respectively. $Y_{0}$ is an open subset in $Y$ containing $Y^{\prime}$. The
intersection $Y_{0}^{\prime}=Y_{0}\cap Y^{\prime}$ is shown as a dark red
line.

Next we define

$\displaystyle\mathcal{D}_{Y_{0}}(\log
Y^{\prime}):=\mathcal{O}_{Y_{0}}\otimes_{\mathcal{O}_{Y}}\mathcal{D}_{Y}(\log
Y^{\prime})\>.$ (36)

The ring structure of $\mathcal{D}_{Y}(\log Y^{\prime})$ induces that of
$\mathcal{D}_{Y_{0}}(\log Y^{\prime})$.

######  Definition 3.2.

A _logarithmic connection_ ${\cal M}$ on $Y_{0}$ along $Y^{\prime}$ is a
$\mathcal{D}_{Y_{0}}(\log Y^{\prime})$-module that is free as an
$\mathcal{O}_{Y_{0}}$-module. More concretely, ${\cal M}$ is isomorphic to
$\mathcal{O}_{Y_{0}}^{r}$ as an $\mathcal{O}_{Y_{0}}$-module for some integer
$r$, and the action of $\partial_{i}$ is given by an $r\times r$ matrix
$P_{i}(z)$ with entries in $\mathcal{O}_{Y_{0}}$ by the formulas

$\displaystyle z_{1}\partial_{1}\,e_{j}$
$\displaystyle=\sum_{k=1}^{r}p_{1j}^{k}(z)\,e_{k}$ (37)
$\displaystyle\partial_{i}\,e_{j}$
$\displaystyle=\sum_{k=1}^{r}p_{ij}^{k}(z)\,e_{k}\>,\quad i=2,\dots,m$ (38)

where $\\{e_{1},\dots,e_{r}\\}$ is some $\mathcal{O}_{Y_{0}}$-free basis.

Setting

$\displaystyle z^{\prime}=(z_{2},\ldots,z_{m})\>,$ (39)

the residue matrix of ${\cal M}$ along $Y^{\prime}$ is given by
$P_{1}(0,z^{\prime})=\bigl{(}p_{1j}^{k}(0,z^{\prime})\bigr{)}_{j,k=1}^{r}\>$.
The residue matrix with respect to a different free basis is obtained by a
conjugation of $P_{1}(0,z^{\prime})\>$. Therefore, matrix multiplication by
$P_{1}(0,z^{\prime})$ from the left is a well-defined linear operator on
$\mathcal{O}_{Y_{0}^{\prime}}\otimes_{\mathcal{O}_{Y_{0}}}{\cal
M}\simeq\mathcal{O}_{Y_{0}^{\prime}}^{r}$, as it does not depend on a
particular choice of the free basis. Given a logarithmic connection ${\cal M}$
on $Y_{0}$ along $Y^{\prime}$ we associate to it a $\mathcal{D}_{Y}$-module
$\mathcal{D}_{Y}\otimes_{\mathcal{D}_{Y}(\log Y^{\prime})}\mathcal{M}\>$,
which is again denoted by the symbol ${\cal M}$ when there is no fear of
confusion. As an $\mathcal{R}$-module, ${\cal M}$ corresponds to a Pfaffian
system (19) with

$\displaystyle P_{1}\ \ \text{replaced by}\ \ \frac{1}{z_{1}}P_{1}(z)\,,$ (40)

where the factor of $1/z_{1}$ comes from (37).

An integrable connection ${\cal M}$ on $Y_{0}\setminus Y^{\prime}$ is said to
be _regular_ along $Y^{\prime}$ if there exists a logarithmic connection
$\widetilde{{\cal M}}$ on $Y_{0}$ along $Y^{\prime}$ whose restriction onto
$Y_{0}\setminus Y^{\prime}$ is ${\cal M}$. This terminology is compatible with
the regular holonomicity of a $\mathcal{D}_{Y}$-module in the following sense:
if ${\cal M}$ is a regular holonomic $\mathcal{D}_{Y}$-module and ${\cal
M}|_{Y_{0}\setminus Y^{\prime}}$ is an integrable connection, then ${\cal M}$
is regular. Although the notion of regularity is well-defined,
$\widetilde{{\cal M}}$ is not uniquely determined by ${\cal M}$. Let us
finally mention that the regularity of $\mathcal{M}$ is equivalent to the
corresponding Pfaffian matrices taking the form (40) after a gauge
transformation.

###  3.2 Restriction of a logarithmic connection in normal form

A matrix is called _resonant_ 888 This is different from resonance in GKZ
systems.  when it has two distinct eigenvalues with an integral difference.
When ${\cal M}$ is a regular connection along $Y^{\prime}$ defined on
$Y_{0}\setminus Y^{\prime}$, there exists a logarithmic connection
$\widetilde{{\cal M}}$ along $Y^{\prime}$ on $Y_{0}$ such that

  1. 1.

The restriction of $\widetilde{{\cal M}}$ onto $Y_{0}\setminus Y^{\prime}$ is
${\cal M}$,

  2. 2.

The residue matrix $P_{1}(0,z^{\prime})$ is non-resonant and the only integral
eigenvalue, if it exists, is $0$.

We say that such an $\widetilde{{\cal M}}$ is in _normal form_. If
$Y^{\prime}=\\{z_{1}=0\\}$, the normal form Pfaffian matrices associated to
$\widetilde{{\cal M}}$ will expand as

$\displaystyle\frac{P_{1}}{z_{1}}$
$\displaystyle=\sum_{n=-1}^{\infty}P_{1,n}(z^{\prime})\,z_{1}^{n}$ (41)
$\displaystyle P_{i}$
$\displaystyle=\sum_{n=0}^{\infty}P_{i,n}(z^{\prime})\,z_{1}^{n}\quad,\quad
i=2,\ldots,m\,.$ (42)

In other words, $P_{1}$ has a logarithmic singularity on $Y^{\prime}$, and the
other Pfaffian matrices are finite there (see Appendix B for more details and
proofs).

The existence of a normal form of a logarithmic connection is well-known
(Deligne, , Proposition 5.4). Working at the level of Pfaffian matrices,
normal form can be found via the standard Moser reduction algorithms of
Barkatou-Jaroschek-Maddah-2017 ; Barkatou-1997 , which we review in Appendix
B.

The next proposition follows from Definition 3.2:

######  Proposition 3.3.

Let ${\cal M}$ be a logarithmic connection along $Y^{\prime}$ on $Y_{0}$. Let
$\\{e_{1},\dots,e_{r}\\}$ be a free basis of ${\cal M}$ and let $P_{i}(z)$ be
the matrices determined by (37) and (38). Let $\mathcal{N}$ be the left
$\mathcal{D}_{Y_{0}}$-submodule of $\mathcal{D}^{r}_{Y_{0}}$ generated by the
row vectors of $\mathbb{1}\,z_{1}\partial_{1}-P_{1}(z)$ and
$\mathbb{1}\,\partial_{i}-P_{i}(z)$ for $i=2,\dots,m$, with $\mathbb{1}$ being
the identity matrix. Then the left $\mathcal{D}_{Y_{0}}$-module ${\cal M}$ is
isomorphic to $\mathcal{D}_{Y_{0}}^{r}/\mathcal{N}$.

Let ${\cal M}$ be a logarithmic connection on $Y_{0}$ along $Y^{\prime}$, and
suppose non-resonance, so that none of the eigenvalues of the residue matrix
$P_{1}(0,z^{\prime})$ is a positive integer. The restriction $\iota^{*}{\cal
M}$ as an integrable connection can be calculated as follows999 Recall that
$\iota$ is the natural inclusion introduced in Section 2.4. . We fix a free
basis $\\{e_{1},\dots,e_{r}\\}$ of ${\cal M}$ and identify it with
$\mathcal{O}_{Y_{0}}^{r}$ as an $\mathcal{O}_{Y_{0}}$-module. We set

$\displaystyle{{\cal
M}}^{\prime}:=\mathcal{O}_{Y_{0}^{\prime}}^{r}\,/\,\mathcal{N}^{\prime}\>,$
(43)

where $\mathcal{N}^{\prime}$ denotes an
$\mathcal{O}_{Y_{0}^{\prime}}$-submodule of $\mathcal{O}_{Y_{0}^{\prime}}^{r}$
spanned by the row vectors of $P_{1}(0,z^{\prime})\>$. From the integrability
condition (18), we have

$\displaystyle\partial_{i}\,P_{1}(0,z^{\prime})=\bigl{[}P_{i}(0,z^{\prime}),\,P_{1}(0,z^{\prime})\bigr{]}\>,\quad\text{for
$i=2,\ldots,m$}\>.$ (44)

It follows easily that the action of $\mathcal{D}_{Y_{0}^{\prime}}$ on
$\mathcal{O}_{Y_{0}^{\prime}}^{r}$ given by

$\displaystyle\partial_{i}\,e_{j}=\sum_{k=1}^{r}p_{ij}^{k}(0,z^{\prime})\,e_{k}$
(45)

induces an action on the quotient ${{\cal M}}^{\prime}\>$.

######  Proposition 3.4.

The restriction $\iota^{*}{\cal M}$ is isomorphic to the integrable connection
${\cal M}^{\prime}$ as a $\mathcal{D}_{Y_{0}^{\prime}}$-module.

See the proof in Appendix D.5.

As the rank of the residue matrix $P_{1}(0,z^{\prime})$ is constant on
$Y_{0}^{\prime}\>$, ${{\cal M}}^{\prime}$ is free as an
$\mathcal{O}_{Y_{0}^{\prime}}$-module (with $Y_{0}^{\prime}$ being replaced by
a smaller Zariski open subset if necessary), so we may denote by
$Q_{i}(z^{\prime})$ the Pfaffian matrices associated to this integrable
connection. In what follows, we give a procedure for finding the Pfaffian
matrices $Q_{i}(z^{\prime})$ associated to the restriction module ${\cal
M}^{\prime}$.

Let $r^{\prime}$ be the holonomic rank of ${\cal M}^{\prime}\>$. For an
element $m$ of $\mathcal{O}_{Y_{0}^{\prime}}^{r}$, we write $[{m}]$ for the
equivalence class in $\mathcal{M}^{\prime}$. We suppose the existence of a
special, free basis $\\{f_{1},\dots,f_{r}\\}$ for
$\mathcal{O}_{Y_{0}^{\prime}}^{r}$ with associated Pfaffian matrices
$\widetilde{Q}_{2}(z^{\prime}),\ldots,\widetilde{Q}_{m}(z^{\prime})$. This
basis is assumed to have the following special property:
$\\{f_{r^{\prime}+1},\dots,f_{r}\\}$ is a basis of $\mathcal{N}^{\prime}$. The
Pfaffian matrix $Q_{i}(z^{\prime})$ of ${\cal M}^{\prime}$ with respect to the
basis $\\{[{f}_{1}],\dots,[{f}_{r^{\prime}}]\\}$ is then given by the first
$r^{\prime}\times r^{\prime}$ block of the matrix
$\widetilde{Q}_{i}(z^{\prime})$.

In general, our given basis $e_{i}$ with associated Pfaffian matrices
$P_{2}(0,z^{\prime}),\ldots,P_{m}(0,z^{\prime})$ will not automatically yield
this block matrix structure. In other words, we know $e_{i}$ and
$P_{i}(0,z^{\prime})$ but we do not know the special basis $f_{i}$ and
$\widetilde{Q}_{i}(z^{\prime})$. Let us thus proceed to construct the gauge
transformation mapping $e_{i}$ to the special basis $f_{i}$.

Given our free basis $\\{e_{1},\dots,e_{r}\\}$ of
$\mathcal{O}_{Y_{0}^{\prime}}^{r}$, we can always find an $r^{\prime}\times r$
matrix $B=(b_{ij})$ with entries in $\mathcal{O}_{Y_{0}^{\prime}}\>$ such that
$\\{[f_{1}],\dots,[f_{r^{\prime}}]\\}$ with $f_{i}=\sum_{j=1}^{r}b_{ij}e_{j}$
forms a free basis of ${\cal M}^{\prime}\>$. Then, by the definition of ${\cal
M}^{\prime}$, there are exactly $r-r^{\prime}$ linearly independent rows of
the residue matrix $P_{1}(0,z^{\prime})\>$. Let $R$ be the matrix containing
these independent rows. Writing
$R=(r_{ij})_{i=r^{\prime}+1,\ldots,r}^{j=1,\ldots,r}$, then
$f_{i}=\sum_{j=1}^{r}r_{ij}e_{j}$ forms a free basis of
$\mathcal{N}^{\prime}$. Aligning the rows of $B$ and $R$ produces an
invertible $r\times r$ matrix $M$:

$\displaystyle
M=\left[\begin{array}[]{c}{\color[rgb]{0.57421875,0.09375,0.09375}B}\\\
\hline\cr{\color[rgb]{0.32421875,0.015625,0.4765625}R}\end{array}\right]=\vbox{\hbox{
\parbox[c]{0.0pt}{\includegraphics{figures/full-mat-brace}} }}\>.$ (48)

In vector notation, we now have $f=M\cdot e$.

Borrowing notation from (28), we thus obtain the Pfaffian matrix
$\widetilde{Q}_{i}(z^{\prime})$ associated to $\\{f_{1},\ldots,f_{r}\\}$ from
the known matrices $P_{i}(0,z^{\prime})$ via a gauge transformation by
$M^{-1}$:

$\displaystyle\widetilde{Q}_{i}(z^{\prime})=(M^{-1})\bigl{[}P_{i}(0,z^{\prime})\bigr{]}=\bigl{(}M\cdot
P_{i}(0,z^{\prime})+\partial_{i}\,M\bigr{)}\cdot M^{-1}\>,\quad i=2,\ldots,m.$
(49)

The sought-after Pfaffian matrix $Q_{i}(z^{\prime})$ for the restriction ideal
$\mathcal{M}^{\prime}$, written in the basis $B$, is finally obtained from the
first $r^{\prime}\times r^{\prime}$ block of $\widetilde{Q}_{i}(z^{\prime})$:

$\displaystyle\bigl{(}\partial_{i}M+M\cdot P_{i}(0,z^{\prime})\bigr{)}\cdot
M^{-1}=\left[\begin{array}[]{cc}{\color[rgb]{0.57421875,0.09375,0.09375}Q_{i}(z^{\prime})}&{\color[rgb]{0.57421875,0.09375,0.09375}\star}\\\
{\color[rgb]{0.32421875,0.015625,0.4765625}\mathbb{0}}&{\color[rgb]{0.32421875,0.015625,0.4765625}\star}\end{array}\right].$
(52)

This formula for obtaining $Q_{i}(z^{\prime})$ will be essential for
calculating restrictions of Pfaffian systems in Section 5. Note that the
lower-left block is zero because $\\{f_{r^{\prime}+1},\dots,f_{r}\\}$ is a
free basis of $\mathcal{N}^{\prime}$ (see also Appendix C.2.1 for a matrix-
based derivation of this fact). This $\mathbb{0}$-block can be used as an
intermediate check on practical computations.

Let us conclude this section by stating an assumption used throughout the rest
of this paper. First, recall that on a non-empty Zariski open subset $Y_{0}$
of $Y$, the $\mathcal{D}_{Y_{0}}$-module $M(\beta;Y)|_{Y_{0}}$ is a regular
integrable connection.

> Assumption: Let $\beta$ be generic. The restriction of a normal form of
> $M_{A}(\beta;Y)|_{Y_{0}}$ to $Y^{\prime}$ is isomorphic to
> $M_{A}(\beta;Y^{\prime})$ as an $\mathcal{R}_{Y^{\prime}}$-module.

By Proposition 3.4 and Appendix B, the restriction of a normal form of
$M_{A}(\beta;Y)|_{Y_{0}}$ to $Y^{\prime}$ can be computed by means of linear
algebra. By this assumption, we can then compute the restriction of
$\mathcal{D}$-modules $M_{A}(\beta;Y^{\prime})\simeq\iota^{*}M_{A}(\beta;Y)$
from the data of $\mathcal{R}\otimes_{\mathcal{D}_{Y}}M_{A}(\beta;Y)\>$.

######  Remark 3.5.

Although we are presently unable to prove this assumption, we have verified in
all our examples that the corank of the residue matrix is always identical to
the expected holonomic rank shown in Theorem 2.1. Moreover, we compute the
$Q_{i}$ matrices in various examples and observe that they agree with the
Pfaffian matrices derived independently using IBP software.

###  3.3 Restriction of solutions to a Pfaffian system

We use the same notation as the previous subsection. In this subsection, we
investigate the boundary value problem of Pfaffian systems in normal form.
This can be seen as a singular boundary value problem; see
(haraoka2002integral, , Section 2), takayama1992propagation and references
therein.

##### Holomorphic restriction

Let ${\cal M}$ be a logarithmic connection on $Y_{0}$ along $Y^{\prime}$ and
let $\\{e_{1},\dots,e_{r}\\}$ be its free basis. Take
$\\{\phi_{1},\dots,\phi_{r}\\}$ to be the dual basis to
$\\{e_{1},\dots,e_{r}\\}$, i.e. each $\phi_{i}$ is an
$\mathcal{O}_{Y_{0}}$-linear map from ${\cal M}$ to $\mathcal{O}_{Y_{0}}$
satisfying $\phi_{i}(e_{j})=\delta_{ij}$. By the definition of the action of
$\partial_{i}$ on ${\cal M}$, we can write the space of holomorphic solutions
of $\mathcal{M}$ on an open set $U$ (recall the definition (32)) as

${\rm Hom}_{\mathcal{D}_{Y_{0}}}({\cal
M},\mathcal{O}^{an}(U))=\left\\{\sum_{i=1}^{r}I_{i}(z)\,\phi_{i}\biggm{|}[I_{1}(z),\dots,I_{r}(z)]^{T}\text{
satisfies (\ref{eqn:pfaffian_system}) with
(\ref{eqn:connection_matrix_log})}\right\\}\>.$ (53)

The vector of functions $I(z)=[I_{1}(z),\dots,I_{r}(z)]^{T}$ is a
solution101010 Indeed, equation (53) implies that for any $m\in\mathcal{M}$ we
have the homomorphism property
$\partial_{i}\bigl{(}\sum_{k=1}^{r}I_{k}(z)\,\phi_{k}(m)\bigr{)}=\sum_{k=1}^{r}I_{k}(z)\,\phi_{k}(\partial_{i}m)$.
So if we choose $m=e_{j}\>$, then
$\partial_{i}\bigl{(}\sum_{i=k}^{r}I_{k}(z)\,\phi_{k}(e_{j})\bigr{)}=\sum_{i=k}^{r}I_{k}(z)\,\phi_{k}(\partial_{i}e_{j}),$
resulting in
$\partial_{i}I_{j}(z)=\sum_{k=1}^{r}I_{k}(z)\,p_{ij}^{k}(0,z^{\prime})\>,$
after we use $\phi_{k}(e_{j})=\delta_{kj}$ and (45). Thus we see that the
functions $I_{j}$ indeed satisfy the Pfaffian system (19).  to the Pfaffian
system (19) and it will be associated to Feynman master integrals later on.

Set $U^{\prime}:=Y^{\prime}\cap U$. We derive a realization of ${\rm
Hom}_{\mathcal{D}_{Y_{0}^{\prime}}}({\cal
M}^{\prime},\mathcal{O}^{an}(U^{\prime}))$ similar to (53). Because
$\mathcal{M}$ is logarithmic, we can expand $P_{1}(z)$ as

$P_{1}(z)=P_{1,-1}(z^{\prime})z_{1}^{-1}+P_{1,0}(z^{\prime})+P_{1,1}(z^{\prime})z_{1}+\ldots$
(54)

The column vector
$I(z^{\prime}):=I(0,z^{\prime})=[I_{1}(0,z^{\prime}),\dots,I_{r}(0,z^{\prime})]^{T}$
then satisfies the following system of PDEs:

$\displaystyle\partial_{i}I(z^{\prime})=P_{i,0}(z^{\prime})\cdot
I(z^{\prime})\>,\quad i=2,\dots,m$ (55) $\displaystyle
P_{1,-1}(z^{\prime})\cdot I(z^{\prime})=0\>.$

Thus, we have an identification

${\rm Hom}_{\mathcal{D}_{Y_{0}^{\prime}}}({\cal
M}^{\prime},\mathcal{O}^{an}(U^{\prime}))=\left\\{\sum_{i=1}^{r}I_{i}(0,z^{\prime})\,\phi_{i}\biggm{|}I(z^{\prime})\text{
is subject to (\ref{eqn:restriction_equation})}\right\\}\>.$ (56)

We call (55) the _holomorphic restriction_ of the Pfaffian system
$\partial_{i}I(z)=P_{i}\cdot I(z)$, as it pertains to solution vectors $I(z)$
which are holomorphic at $z_{1}=0$. Note that the first condition involving
the residue matrix $P_{1,-1}(z^{\prime})$ imposes constraints on the solution
vector $I(z^{\prime})$. These extra relations lead to a drop in rank in
comparison to the unrestricted Pfaffian system.

Solutions to Pfaffian systems (40) and solutions of (55) are related to each
other in the following sense:

######  Theorem 3.6.

Suppose that the Pfaffian system (40) is non-resonant. Then there is a one-to-
one correspondence between the solutions $I(z)$ of (40) being holomorphic at
$(z_{1},z^{\prime})=(0,0)$ and the holomorphic solutions $I(z^{\prime})$ of
(55). The correspondence is given by the boundary value map $I(z)\mapsto
I(0,z^{\prime})$.

See the proof in Appendix D.6.

##### Logarithmic restriction

So far, we have only focused on limits of Pfaffian systems for which the
solution $I$ is holomorphic. However, in physics applications, it is
interesting to study solutions that develop logarithmic singularities in the
restriction limit $z_{1}\to 0$. Let us hence define
$\widetilde{\mathcal{O}}^{(N)}$ to be the vector space of functions of the
form

$\displaystyle f_{0}(z)+f_{1}(z)\log z_{1}+\ldots+f_{N}(z)\log^{N}z_{1}\,,$
(57)

where the $f_{i}(z)$ denote convergent power series at the origin. Then we
have an identification

${\rm Hom}_{\mathcal{D}_{Y_{0}}}({\cal
M},\widetilde{\mathcal{O}}^{(N)})=\left\\{\sum_{i=1}^{r}I_{i}(z)\,\phi_{i}\biggm{|}I_{i}(z)\in\widetilde{\mathcal{O}}^{(N)},\,[I_{1},\ldots,I_{r}]^{T}\text{
satisfies (\ref{eqn:pfaffian_system}) with
(\ref{eqn:connection_matrix_log})}\right\\}\>.$ (58)

We fix a free basis $\\{e_{1},\dots,e_{r}\\}$ of ${\cal M}$ to identify it
with $\mathcal{O}_{Y_{0}}^{r}$ as an $\mathcal{O}_{Y_{0}}$-module. For any
non-negative integer $N$, we set ${{\cal
M}}^{\prime}_{N}:=\mathcal{O}_{Y_{0}}^{r}/{\rm
Im}\,P_{1}(0,z^{\prime})^{N+1}$. As discussed in the construction of ${\cal
M}^{\prime}$ in Section 3.2, it follows that the action of
$\mathcal{D}_{Y_{0}^{\prime}}$ on $\mathcal{O}_{Y_{0}}^{r}$ given by (45)
induces that on the quotient ${{\cal M}}^{\prime}$.

Consider now a vector expression of the form

$\displaystyle I^{(0)}(z)+I^{(1)}(z)\log
z_{1}+\ldots+I^{(N)}(z)\log^{N}z_{1}.$ (59)

We present the following generalization of (55), which is a system of PDEs for
the vector functions $I^{(i)}(0,z^{\prime})=I^{(i)}(z^{\prime})$:

$\displaystyle\begin{array}[]{l
l}\partial_{i}\,I^{(n)}(z^{\prime})=P_{i,0}(z^{\prime})\cdot
I^{(n)}(z^{\prime})\>,&\quad i=2,\dots,m,\ n=0,\dots,N\\\\[10.0pt]
P_{1,-1}(z^{\prime})\cdot
I^{(n)}(z^{\prime})=(n+1)I^{(n+1)}(z^{\prime})\>,&\quad
n=0,\ldots,N\end{array}$ (62)

with $I^{(N+1)}(z^{\prime})=0$. We call (62) the _logarithmic restriction_ of
(40) at order $N$. We have an identification

${\rm Hom}_{\mathcal{D}_{Y_{0}^{\prime}}}({\cal
M}^{\prime}_{N},\widetilde{\mathcal{O}}^{(N)})=\left\\{\sum_{i=1}^{r}I^{(i)}(z^{\prime})\,\phi_{i}\biggm{|}I^{(i)}(z^{\prime})\in\mathcal{O}^{an},\,I^{(i)}\text{
is subject to (\ref{eq:log_constraint})}\right\\}\>.$ (63)

######  Theorem 3.7.

Suppose that the Pfaffian system (40) is non-resonant. Then there is a one-to-
one correspondence between the solutions $I(z)$ to (40) of the form
$I^{(0)}(z)+I^{(1)}(z)\log z_{1}+\ldots+I^{(N)}(z)\log^{N}z_{1}$ and the
holomorphic solutions $I^{(n)}(z^{\prime})$ of (62). More explicitly, the
correspondence is given by the boundary value map $I(z)\mapsto
I^{(0)}(0,z^{\prime})$.

See the proof in Appendix D.7.

###  3.4 Relation to Feynman integrals

Let us summarize how the preceding sections relate to the study of Feynman
integrals.

Fix a single Feynman integral. Its associated de Rham cohomology group
$M_{A}(\beta;Y)$ (see (10)) carries the structure of a holonomic
$\mathcal{D}_{Y}$-module, so the previous results on $\mathcal{D}$-modules and
Pfaffian systems also apply to $M_{A}(\beta;Y)$. In particular, there exist an
$r$-dimensional vector of master integrals $I=I(z)$ and Pfaffian matrices
$P_{i}(z)$ such that $\partial_{i}I=P_{i}(z)\cdot I(z)$ holds for
$i=1,\ldots,m$. Assume that this Pfaffian system has a singularity at
$z_{1}=0$. We recall the notation $z^{\prime}=(0,z_{2},\ldots,z_{m})$.

According to the discussion of Section 3.2, there exists a normal form of
$M_{A}(\beta;Y)$. In practice, this means that we can use Moser reduction to
find a set of new Pfaffian matrices $\widetilde{P}_{i}(z)$ such (i)
$\widetilde{P}_{1}(z)$ has a simple pole at $z_{1}=0$, (ii)
$\widetilde{P}_{2}(z_{1}),\ldots,\widetilde{P}_{m}(z)$ are holomorphic along
$\\{z_{1}=0\\}$, and (iii) the spectrum of the residue matrix
$\widetilde{P}_{1,-1}(z^{\prime})$ is non-resonant (i.e., no difference of
eigenvalues is integral) with zero being the unique integral eigenvalue. By
abuse of notation, we keep using the notation $I$ for the solution vector in
this new basis.

We are interested in a system of _simpler_ PDEs which holds in the limit
$z_{1}\to 0$. This problem is split into two cases: (i) the solution vector
$I$ is finite at $z_{1}=0$, or (ii) the solution vector is $\log$-singular at
$z_{1}=0$.

##### Case (i): Holomorphic restriction

The relevant set of PDEs was given in (55). However, this is written in a
redundant fashion because we have not resolved the constraint111111This
constraint can be thought of as a collection of IBP relations which hold in
the limit.

$\displaystyle\widetilde{P}_{1,-1}\cdot I(0,z^{\prime})=0\,,$ (64)

where the limit $z_{1}\to 0$ on $I(z_{1},z^{\prime})$ is taken at the
_integrand_ level. There are
$\texttt{Rank[}\,\widetilde{P}_{1,-1}\,\texttt{]}$ many relations above, so we
would like a restricted PDE system which is manifestly of dimension
$r^{\prime}=r-\texttt{Rank[}\,\widetilde{P}_{1,-1}\,\texttt{]}$. Assume we
know an $r^{\prime}$-dimensional basis $J(z^{\prime})$ for the restricted
system (in Section 5.2 and Appendix C.2.2, we show how to construct such a
basis from the original, $r$-dimensional one). This basis fixes the matrix $B$
in (48) according to the relation $J=B\cdot I(0,z^{\prime})$. Moreover, the
matrix $R$ in (48) equals
$\texttt{RowReduce[}\,\widetilde{P}_{1,-1}\,\texttt{]}$, where the operation
RowReduce includes the deletion of zero-rows. So we now know the matrix $M$
from that same formula. Equation (52) finally yields the $r^{\prime}\times
r^{\prime}$-dimensional Pfaffian matrices $Q_{i}(z^{\prime})$ satisfying

$\displaystyle\partial_{i}J=Q_{i}\cdot J\quad,\quad i=2,\ldots,m.$ (65)

This restricted Pfaffian system is simpler than the original one in the sense
that it has a smaller rank, and it depends on one less variable. If needed,
once $J$ has been determined, it is possible to recover $I$ by inverting the
relation

$\displaystyle M\cdot I(0,z^{\prime})=\left[\begin{array}[]{c}B\cdot
I(0,z^{\prime})\\\ \hline\cr R\cdot
I(0,z^{\prime})\end{array}\right]=\left[\begin{array}[]{c}J\\\ \hline\cr
0\\\\[-5.0pt] \vdots\\\ 0\end{array}\right]\,,$ (72)

where the block of $r-r^{\prime}$ zeros below $J$ on the RHS arises due to
(64).

Let us summarize these steps in an algorithm.

######  Algorithm 3.8.

(Pfaffian-level holomorphic restriction to $z_{1}=0$)
Input: The $r$-dimensional Pfaffian system $\partial_{i}I(z)=P_{i}(z)\cdot
I(z),\,i=1,\ldots m$.
Output: The solution vector $I(0,z^{\prime})$.

1:Gauge transform $P_{i}\to\widetilde{P}_{i}:=G[P_{i}]$ to _normal form_ via
Moser reduction.

2:Find the residue matrix $\widetilde{P}_{1,-1}$ and its row-reduced form
$R:=\texttt{RowReduce[}\,\widetilde{P}_{1,-1}\,\texttt{]}$. Set
$r^{\prime}=r-\texttt{Rank[}\,R\,\texttt{]}$.

3:Choose an $r^{\prime}$-dimensional basis $J$. This yields the
$r^{\prime}\times r$-dimensional matrix $B$ via $J=B\cdot I(0,z^{\prime})$. If
there is no obvious choice for $J$, then $B$ can be set to
$B=\texttt{NullSpace[}\,R\,\texttt{]}$.

4:Construct the invertible block matrix $M=\left[\begin{array}[]{c}B\\\
\hline\cr R\end{array}\right]$.

5:For each $i=2,\ldots,m$, compute $\bigl{(}\partial_{i}M+M\cdot
P_{i}(0,z^{\prime})\bigr{)}\cdot M^{-1}$ and save the upper-left
$r^{\prime}\times r^{\prime}$ block matrices $Q_{i}(z^{\prime})$.

6:Solve the Pfaffian system $\partial_{i}J(z^{\prime})=Q_{i}(z^{\prime})\cdot
J(z^{\prime})$ for $J$.

7:return $I(0,z^{\prime})=M^{-1}\cdot[J\,|\,0\cdots 0\,]^{T}$.

We visualize this algorithm in Figure 2 by showing the building blocks of the
restricted Pfaffian system and its solution introduced in Section 3.2. The red
surface depicts every possible solution $I(0,z_{2})$ which satisfies the
constraint $\widetilde{P}_{1,-1}(z_{2})\cdot I(0,z_{2})=0$, i.e. the nullspace
of $\widetilde{P}_{1,-1}$. For a fixed value of $z_{2}$, this nullspace spans
a $1$-dimensional vector space, shown as red lines. As $z_{2}$ varies, these
lines trace out the red surface. This surface thus portrays a basis for the
restricted Pfaffian system, as given by the row-span of the
${\color[rgb]{0.57421875,0.09375,0.09375}B}$ matrix in (48). For fixed
$z_{2}$, the purple lines represent the $1$-dimensional span of the row
vectors in $\widetilde{P}_{1,-1}$. As $z_{2}$ varies, we obtain the purple
surface. This surface hence depicts the
${\color[rgb]{0.32421875,0.015625,0.4765625}R}$ matrix in (48). Orthogonality
of the purple surface w.r.t. the red surface ensures invertibility of the $M$
matrix in (48). The yellow curve illustrates a _particular_ solution vector
$I(0,z_{2}):=[I_{1}(0,z_{2}),I_{2}(0,z_{2})]^{T}\>$ given some initial
condition $I(0,0)$. Note that this solution is constrained to lie on the red
surface.

Figure 2:  Algorithm 3.8 exemplified as the restriction $z_{1}\to 0$ of a rank
$r=2$ Pfaffian system in two variables $z_{1}$ and $z_{2}$.

##### Case (ii): Logarithmic restriction

By the general theory of PDEs haraoka2020linear , we expect the solution
vector $I(z)$ to take the following form in the limit $z_{1}\to 0$:

$\displaystyle
I^{(\log)}(z)\quad=\quad\sum_{\lambda\,\in\,\texttt{Spec[}\,\widetilde{P}_{1,-1}\,\texttt{]}}z_{1}^{\lambda}\hskip
5.69046pt\sum_{n=0}^{N_{\lambda}}I^{(\lambda,n)}(z^{\prime})\log^{n}(z_{1}).$
(73)

The outer sum runs over the unique eigenvalues $\lambda$ of the residue matrix
$\widetilde{P}_{1,-1}$, and for each $\lambda$ there is a collection of
$N_{\lambda}+1$ unknown vectors
$I^{(\lambda,0)},\ldots,I^{(\lambda,N_{\lambda})}$. The integer $N_{\lambda}$
depends on the structure of the Jordan blocks in the Jordan decomposition of
$\widetilde{P}_{1,-1}$.

For fixed $\lambda,$ each $I^{(\lambda,n)}$ is subject to the system of PDEs
(62). This system includes a list of constraints

$\displaystyle(\widetilde{P}_{1,-1}-\lambda)\cdot
I^{(\lambda,n)}=(n+1)I^{(\lambda,n+1)}\quad,\quad n=1,\ldots,N_{\lambda},$
(74)

which leads to a drop in rank. Equation (62) was specifically written for the
eigenvalue $\lambda=0$, but in the general case there is a shift by $-\lambda$
as on the LHS. We observe that the constraints above imply

$\displaystyle
I^{(\lambda,n)}=(n+1)!\,(\widetilde{P}_{1,-1}-\lambda)^{n+1}\cdot
I^{(\lambda,0)}\quad,\quad n=1,\ldots,N_{\lambda},$ (75)

for which reason we only need to compute $I^{(\lambda,0)}$. Setting
$n=N_{\lambda}$ above, we have

$\displaystyle(\widetilde{P}_{1,-1}-\lambda)^{N_{\lambda}+1}\cdot
I^{(\lambda,0)}=0\,,$ (76)

giving $\texttt{Rank[}\,\widetilde{P}_{1,-1}^{N_{\lambda}+1}\,\texttt{]}$ many
constraints among the entries of the solution vector $I^{(\lambda,0)}$.

For each $\lambda$, we calculate $I^{(\lambda,0)}$ by repeating the procedure
from case (i). The first step is to find an $r^{\prime}_{\lambda}$-dimensional
basis $J^{(\lambda)}$, where
$r^{\prime}_{\lambda}=r-\texttt{Rank[}\,(\widetilde{P}_{1,-1}-\lambda)^{N_{\lambda}+1}\,\texttt{]}$,
so as to fix the matrix $B^{(\lambda)}$ in

$\displaystyle M^{(\lambda)}=\left[\begin{array}[]{c}B^{(\lambda)}\\\
\hline\cr R^{(\lambda)}\end{array}\right].$ (79)

It is important that $B^{(\lambda)}$ be chosen such that $M^{(\lambda)}$ is
invertible. The $R^{(\lambda)}$ block encodes the constraints from the residue
matrix in (76):

$\displaystyle
R^{(\lambda)}=\texttt{RowReduce[}\,(\widetilde{P}_{1,-1}-\lambda)^{N_{\lambda}+1}\,\texttt{]}\,.$
(80)

Because of (76), we thus get a block of zeros in

$\displaystyle M^{(\lambda)}\cdot
I^{(\lambda,0)}=\left[\begin{array}[]{c}B^{(\lambda)}\cdot I^{(\lambda,0)}\\\
\hline\cr R^{(\lambda)}\cdot
I^{(\lambda,0)}\end{array}\right]=\left[\begin{array}[]{c}J^{(\lambda)}\\\
\hline\cr 0\\\\[-5.0pt] \vdots\\\ 0\end{array}\right]\ .$ (87)

This can be inverted to recover $I^{(\lambda,0)}$ once we have computed
$J^{(\lambda)}$. Inserting $M^{(\lambda)}$ into (52), we obtain a collection
of $r^{\prime}_{\lambda}\times r^{\prime}_{\lambda}$-dimensional Pfaffian
matrices $Q^{(\lambda)}_{i}(z^{\prime})$ such that

$\displaystyle\partial_{i}J^{(\lambda)}=Q^{(\lambda)}_{i}\cdot
J^{(\lambda)}\quad,\quad i=2,\ldots,m.$ (88)

The equations (75, 87, 88) together determine the unknown vectors
$I^{(\lambda,n)}$ in (73), apart from fixing the boundary constants stemming
from solutions of the Pfaffian system (88). These constants can e.g. be fixed
by comparing with the original integral $I(z)$ at special numerical points.

Summarizing all the steps above, we have

######  Algorithm 3.9.

(Pfaffian-level logarithmic restriction to $z_{1}=0$)
Input: The $r$-dimensional Pfaffian system $\partial_{i}I(z)=P_{i}(z)\cdot
I(z),\,i=1,\ldots m$.
Output: The solution vector $I^{(\log)}(z)$.

1:Gauge transform $P_{i}\to\widetilde{P}_{i}:=G[P_{i}]$ to _normal form_ via
Moser reduction.

2:Find the residue matrix $\widetilde{P}_{1,-1}$.

3:for each unique $\lambda\in\texttt{Spec[}\,\widetilde{P}_{1,-1}\,\texttt{]}$
do

4: Compute
$R^{(\lambda)}=\texttt{RowReduce[}\,(\widetilde{P}_{1,-1}-\lambda)^{N_{\lambda}+1}\,\texttt{]}$
and set $r^{\prime}_{\lambda}=r-\texttt{Rank[}\,R^{(\lambda)}\,\texttt{]}$.

5: Choose an $r^{\prime}_{\lambda}$-dimensional basis $J^{(\lambda)}$. This
fixes an $r^{\prime}_{\lambda}\times r$-dimensional matrix $B^{(\lambda)}$ via
$J^{(\lambda)}=B^{(\lambda)}\cdot I^{(\lambda,0)}$. One choice is
$B^{(\lambda)}=\texttt{NullSpace[}\,R^{(\lambda)}\,\texttt{]}$.

6: Construct the invertible block matrix
$M^{(\lambda)}=\left[\begin{array}[]{c}B^{(\lambda)}\\\ \hline\cr
R^{(\lambda)}\end{array}\right]$.

7: For each $i=2,\ldots,m$, compute
$\bigl{(}\partial_{i}M^{(\lambda)}+M^{(\lambda)}\cdot
P_{i}(0,z^{\prime})\bigr{)}\cdot(M^{(\lambda)})^{-1}$ and save the upper-left
$r^{\prime}_{\lambda}\times r^{\prime}_{\lambda}$ block matrices
$Q^{(\lambda)}_{i}(z^{\prime})$.

8: Solve the Pfaffian system
$\partial_{i}J^{(\lambda)}(z^{\prime})=Q_{i}^{(\lambda)}(z^{\prime})\cdot
J^{(\lambda)}(z^{\prime})$ for $J^{(\lambda)}$.

9: Compute
$I^{(\lambda,0)}(z^{\prime})=(M^{(\lambda)})^{-1}\cdot[J^{(\lambda)}\,|\,0\cdots
0\,]^{T}$

10: Compute
$I^{(\lambda,n)}(z^{\prime})=(n+1)!\,(\widetilde{P}_{1,-1}-\lambda)^{n+1}\cdot
I^{(\lambda,0)}$ for each $n=1,\ldots,N_{\lambda}$.

11:end for

12:return The sum (73).

This is a novel approach for computing logarithmic corrections to scattering
events. A pedagogical example is given in Section 5.2, where we compute
logarithmic corrections to a one-loop box diagram contributing to Bhabha
scattering. In that example we imagine that $z_{1}$ denotes a small mass, such
as that of the electron. While the goal of the example is to compute
$I^{(\lambda,\bullet)}(z^{\prime})$, the actual, full, solution $I(z)$ would
read

$\displaystyle I(z)\ \
=\sum_{\lambda\,\in\,\texttt{Spec[}\,\widetilde{P}_{1,-1}\,\texttt{]}}z_{1}^{\lambda}\hskip
5.69046pt\sum_{n=0}^{N_{\lambda}}I^{(\lambda,n)}(z_{1},z^{\prime})\log^{n}(z_{1})\,,$
(89)

where $I^{(\lambda,n)}(z_{1},z^{\prime})$ is the analytic function satisfying
$I^{(\lambda,n)}(0,z^{\prime})=I^{(\lambda,n)}(z^{\prime})$. Using the
recursion relations (516, 527) from the proofs of Theorem 3.6 and Theorem 3.7,
presented in Appendix D, it it possible to reconstruct
$I^{(\lambda,n)}(z_{1},z^{\prime})$ from $I^{(\lambda,n)}(z^{\prime})$ as a
power series in $z_{1}$. In other words, one would thereby obtain terms of the
form $z_{1}^{a}\log^{b}(z_{1})$ for $a,b\in\mathbb{Z}_{>0}$. Explicit
calculations of this nature are left for future work.

##  4 Restrictions as $\mathcal{D}$-modules by the Macaulay matrix method

In this section, we will discuss rational restrictions from the
$\mathcal{D}$-module point of view. The first subsection gives an algorithm
for finding bases of rational restrictions. The second subsection uses these
bases to construct rational restrictions via the Macaulay matrix method.

###  4.1 Restrictions as ${\cal D}$-modules

Recall the notion of rational restrictions from Definition 2.11. There, $Y$ is
the $m$-dimensional complex plane and $Y^{\prime}$ is
$(m-m^{\prime})$-dimensional complex plane in $Y$ defined by
$z_{1}=\ldots=z_{m^{\prime}}=0$. The following proposition will be used to
prove the correctness of Algorithm C.2 later. A proof is given in Appendix
D.8.

######  Proposition 4.1.

Let $\mathcal{I}$ be a regular holonomic ideal of ${\cal D}_{Y}$. We have the
isomorphism of left ${\cal R}_{Y^{\prime}}$-modules

${\cal R}_{Y^{\prime}}\otimes_{{\cal D}_{Y^{\prime}}}{\cal
D}_{Y}/(\mathcal{I}+\sum_{j=1}^{m^{\prime}}z_{j}{\cal D}_{Y})\simeq{\cal
R}_{Y^{\prime}}\otimes_{{\cal D}_{Y^{\prime}}}{\cal D}_{Y}/(({\cal
R}_{Y}\mathcal{I})\cap{\cal D}_{Y}+\sum_{j=1}^{m^{\prime}}z_{j}{\cal
D}_{Y})\,.$ (90)

The ideal $({\cal R}_{Y}\mathcal{I})\cap{\cal D}_{Y}$ is known as the Weyl
closure of $\mathcal{I}$ (Tsai2001algorithms, , Definition 5.1).

Let $\mathcal{I}$ be a left holonomic ideal in ${\cal D}_{Y}$. The restriction
of the left holonomic ${\cal D}_{Y}$-module ${\cal M}={\cal
D}_{Y}/\mathcal{I}$ to $Y^{\prime}$ is defined by $\iota^{*}{\cal M}={\cal
D}_{Y}/(\mathcal{I}+z_{1}{\cal D}_{Y}+\ldots+z_{m^{\prime}}{\cal D}_{Y})$. We
denote the restriction $\iota^{*}{\cal M}$ by ${\cal N}_{Y^{\prime}}$. When
$m=m^{\prime}$, we denote the restriction by ${\cal N}_{0}$.

Let $w\in\mathbb{Z}^{m}$ be a vector and let $w_{i}$ denote its $i$-th
component. Define the $(-w,w)$-_order_ for $cz^{\alpha}\partial^{\beta}$,
$c\in\mathbb{C}$, by

$\mathrm{ord}\,_{(-w,w)}(cz^{\alpha}\partial^{\beta})=-w\cdot\alpha+w\cdot\beta\,,$
(91)

where $z^{\alpha}=\prod_{i=1}^{m}z_{i}^{\alpha_{i}}$,
$\partial^{\beta}=\prod_{i=1}^{m}\partial_{i}^{\beta_{i}}$ and $\cdot$ is the
standard inner product. The order is defined for $\ell\in{\cal D}_{Y}$ as the
maximum of all monomial $(-w,w)$-orders appearing in $\ell$. For
$\ell,\ell^{\prime}\in{\cal D}_{Y}$, we have
$\mathrm{ord}\,_{(-w,w)}(\ell\ell^{\prime})=\mathrm{ord}\,_{(-w,w)}(\ell)+\mathrm{ord}\,_{(-w,w)}(\ell^{\prime})$.
Putting $\theta_{i}=z_{i}\partial_{i}$, we note that
$\mathrm{ord}\,_{(-w,w)}(\theta_{i})=0$. Since $w$ defines the order among
monomials, it is called the _weight vector_.

Now, let us briefly review an algorithm for constructing the restriction
${\cal N}_{Y^{\prime}}$ of ${\cal M}$ by following the exposition (SST, ,
Section 5.2). Let $w\in\mathbb{Z}^{m}$ be a weight vector such that

$w_{m^{\prime}+1}=\ldots=w_{m}=0\quad,\quad w_{i}>0\quad,\quad 1\leq i\leq
m^{\prime}.$ (92)

Let $b(u)$ be the $b$-_function_ (also called the indicial polynomial) of the
left ideal $\mathcal{I}$ in ${\cal D}_{Y}$ with respect to $w$. More
precisely, $b(\sum_{i=1}^{m^{\prime}}w_{i}z_{i}\partial_{i})$ is a monic
generator of the principal ideal

$\mathbb{C}\left[\sum_{i=1}^{m^{\prime}}w_{i}z_{i}\partial_{i}\right]\cap{\rm
in}_{(-w,w)}\mathcal{I}\,,$ (93)

where ${\rm in}_{(-w,w)}(f)$ is the sum of the highest order terms in
$f\in{\cal D}_{Y}$ such that the weight for $z_{i}$ is $-w_{i}$ and the weight
for $\partial_{i}$ is $w_{i}$ (SST, , Section 1.1)121212 We note that several
computer algebra systems have commands for computing $b$-functions - see in
particular (SST, , p. 194) for a definition of the $b$-function which is
computer algebra friendly and dojo for details on computer algebra systems
which are useful in this context. .

Fix a weight vector $w$ satisfying (92). Let $\mathcal{F}_{k}$ be the set of
all elements in ${\cal D}_{Y}$ such that the $(-w,w)$-order is less than or
equal to $k$. In other words, we have

$\mathcal{F}_{k}=\left\\{\sum
c_{\alpha\beta}z^{\alpha}\partial^{\beta}\Bigm{|}-w\cdot\alpha+w\cdot\beta\leq
k\right\\}\>,$ (94)

which is a free left ${\cal D}_{Y^{\prime}}$-module.

######  Theorem 4.2.

Oaku-1997 , (SST, , Section 5.2)

  1. 1.

If $b(u)=0$ does not have a non-negative integral root, then the restriction
${\cal N}_{Y^{\prime}}=0$.

  2. 2.

Let $u_{0}$ be the maximal non-negative integral root of $b(u)=0$ and let
$\\{g_{1},\ldots,g_{t}\\}$ be a $(-w,w)$-Gröbner basis of $\mathcal{I}$ (see
(SST, , Section 1.1)). Then the restriction ${\cal N}_{Y^{\prime}}$ is
expressed as

${\cal
N}_{Y^{\prime}}\simeq\frac{\mathcal{F}_{u_{0}}}{\sum_{\\{i\in\\{1,\ldots,t\\}\,|\,\mathrm{ord}\,_{(-w,w)}g_{i}\leq
u_{0}\\}}\mathcal{F}_{u_{0}-\mathrm{ord}\,_{(-w,w)}(g_{i})}g_{i}+\sum_{j=1}^{m^{\prime}}z_{j}\mathcal{F}_{u_{0}+w_{j}}}$
(95)

as a left ${\cal D}_{Y^{\prime}}$-module.

The right hand side of (95) can also be expressed as

$\frac{\sum_{w\cdot\alpha\leq u_{0}}{\cal
D}_{Y^{\prime}}\prod_{j=1}^{m^{\prime}}\partial_{j}^{\alpha_{j}}}{\sum_{\\{i\in\\{1,\ldots,t\\}\,|\,\mathrm{ord}\,_{(-w,w)}g_{i}\leq
u_{0}\\}}{\cal D}_{Y^{\prime}}\sum_{w\cdot\beta\leq
u_{0}-\mathrm{ord}\,_{(-w,w)}(g_{i})}:\prod_{j=1}^{m^{\prime}}\partial_{j}^{\beta_{j}}g_{i}:{|_{z_{1}=\ldots=z_{m^{\prime}}=0}}}\,.$
(96)

The notation $:\ell:{|_{z_{1}=\ldots=z_{m^{\prime}}=0}}$ means that we first
order each monomial in $\ell$ such that $\partial_{i}$’s come to the right and
$z_{j}$’s come to the left (via $[\partial_{i},z_{j}]=\delta_{ij}$),
whereafter we set $z_{1}=\ldots=z_{m^{\prime}}=0$. For example, we have

$:\partial_{1}z_{1}:\big{|}_{z_{1}=0}=\>\>(z_{1}\partial_{1}+1)\big{|}_{z_{1}=0}=1.$

Note that $u_{0}$ may be set equal to a non-negative integer larger than or
equal to the maximal non-negative integral root of $b(u)=0$. This procedure
will be utilized in the following sections.

######  Remark 4.3.

Let us regard ${\cal R}_{Y^{\prime}}\otimes_{{\cal D}_{Y^{\prime}}}{\cal
N}_{Y^{\prime}}$ as a vector space over
$\mathbb{C}(z_{m^{\prime}+1},\ldots,z_{m})$. Considering a Gröbner basis in a
free ${\cal R}_{Y^{\prime}}$-module for the denominator of the expression
(96), we can take a basis of the vector space consisting only of monomials in
$\partial_{i}$’s.

We proceed by constructing an algorithm for finding the Pfaffian system
associated to the restriction ${\cal N}_{Y^{\prime}}$ via Gröbner bases for
submodules. This algorithm will lend itself to the Macaulay matrix restriction
method presented in Section 4.2 by providing a standard basis.

In order to simplify our presentation, we first illustrate our algorithms for
$m=2$ and $m^{\prime}=1$. The generalization to more than $2$ variables is not
difficult, and will be discussed later. In the rest of this section, we set
$x=z_{1},y=z_{2}$ and ${\cal D}^{\prime}={\cal D}_{m},{\cal D}={\cal
D}_{m-m^{\prime}}$.

We want to restrict a given left ideal $\mathcal{I}={\cal
D}^{\prime}\cdot\\{f_{1},\ldots,f_{\mu}\\}$ in ${\cal
D}^{\prime}=\mathbb{C}\langle x,y,\partial_{x},\partial_{y}\rangle$ to $x=0$.
The restriction module is defined as

${\cal D}^{\prime}/(\mathcal{I}+x{\cal D}^{\prime}).$

As stated in Theorem 2.12, when ${\cal D}^{\prime}/\mathcal{I}$ is a holonomic
${\cal D}^{\prime}$-module, the restriction module is a holonomic ${\cal
D}$-module with ${\cal D}=\mathbb{C}\langle y,\partial_{y}\rangle$.

Let $\mathcal{J}$ be a left submodule of ${\cal D}^{k+1}$. The holonomic rank
of the left ${\cal D}$-module ${\cal D}^{k+1}/\mathcal{J}$ is, by definition,
$\mathrm{dim}_{\mathbb{C}(y)}{\cal R}^{k+1}/({\cal R}\mathcal{J})$, where
${\cal R}=\mathbb{C}(y)\langle\partial_{y}\rangle$.

Let $\ell\in{\cal D}^{\prime}$ and take $k$ to be a sufficiently large number.
We regard the operator

$\ell=\sum_{j=0}^{k}c_{j}(x,y,\partial_{y})\partial_{x}^{j}$

as an element of ${\cal D}[x]^{k+1}$ by

$v_{k}(\ell):=(c_{k},c_{k-1},\ldots,c_{1},c_{0})\in{\cal D}[x]^{k+1}.$ (97)

We regard ${\cal D}[x]^{k+1}$ (resp. ${\cal D}^{k+1}$) as a subset of ${\cal
D}[x]^{k+2}$ (resp. ${\cal D}^{k+2}$) by the inclusion

${\cal
D}[x]^{k+1}\ni(c_{k},c_{k-1},\ldots,c_{1},c_{0})\mapsto(0,c_{k},c_{k-1},\ldots,c_{1},c_{0})\in{\cal
D}[x]^{k+2}.$

Let $u_{0}$ be the maximal non-negative integral root of the $b$-function of
$\mathcal{I}$ for the weight $w=(1,0)$. If there is no such $u_{0}$, the
restriction is $0$. Let $u_{1}$ be the maximal $(-w,w)$-order of the elements
in a $(-w,w)$-Gröbner basis for $\mathcal{I}$ (see (SST, , Section 1.1) for
details on Gröbner bases of Weyl algebras).

######  Algorithm 4.4.

(Rational restriction to $x=0$)
Input: Generators $\\{f_{1},\ldots,f_{\mu}\\}$ of a holonomic ideal
$\mathcal{I}$ in ${\cal D}^{\prime}$. Holonomic rank $r$ of the restriction of
${\cal D}^{\prime}/\mathcal{I}$ to $x=0$. A positive integer $\gamma$ such
that $\gamma\geq\max(u_{0}+1,u_{1}+1)$.
Output: Generators of a left submodule $\mathcal{J}\cap{\cal D}^{\gamma}$ of
${\cal D}^{\gamma}$ such that ${\cal R}^{\gamma}/{\cal R}(\mathcal{J}\cap{\cal
D}^{\gamma})\simeq{\cal R}\otimes_{{\cal D}}{\cal
D}^{\prime}/(\mathcal{I}+x{\cal D}^{\prime})$, that is the rational
restriction.

1:$w=(1,0)$

2:$k=\gamma-1$

3:repeat

4: $\mathcal{J}={\cal
D}\cdot\Bigl{\\{}v_{k}\,\bigl{(}\,:\partial_{x}^{j}f_{i}:{\big{|}_{x=0}}\bigr{)}\Bigm{|}\mathrm{ord}\,_{(-w,w)}(\partial_{x}^{j}f_{i})\leq
k,1\leq i\leq N,\ j\in\mathbb{N}_{0}\Bigr{\\}}$

5: $k:=k+1$.

6:until $\mathrm{rank}\bigl{(}{\cal D}^{\gamma}/\mathcal{J}\cap{\cal
D}^{\gamma}\bigr{)}=r$

7:return $\mathcal{J}\cap{\cal D}^{\gamma}$

We can obtain generators of $\mathcal{J}\cap{\cal D}^{\gamma}$ by the POT
(position over term) order in ${\cal D}^{k+1}$ (see Appendix A for Gröbner
bases in a free module and the POT order). Once we get a set of generators of
$\mathcal{J}\cap{\cal D}^{\gamma}$, the Pfaffian system of the restriction
${\cal D}^{\prime}/(\mathcal{I}+x{\cal D}^{\prime})$ can be obtained from a
Gröbner basis of ${\cal R}(\mathcal{J}\cap{\cal D}^{\gamma})$. An improvement
of this algorithm is given in Appendix C.1.

######  Theorem 4.5.

Algorithm 4.4 stops at some number $k$ and gives the correct answer.

The proof of this theorem is technical; see Appendix D.9.

Let us illustrate Algorithm 4.4 with a small input.

######  Example 4.6.

Let $\mathcal{I}$ be the left ideal generated by

$\displaystyle f_{1}$ $\displaystyle=$ $\displaystyle
2\theta_{x}\theta_{y}-1=2xy\partial_{x}\partial_{y}-1,$ $\displaystyle f_{2}$
$\displaystyle=$ $\displaystyle
2\theta_{y}^{2}+\theta_{x}-\theta_{y}-1=2(y^{2}\partial_{y}^{2}+y\partial_{y})+x\partial_{x}-y\partial_{y}-1.$

The solution space for $\mathcal{I}$ is spanned by
$\\{xy^{1/2},\,x^{1/\sqrt{2}}y^{1/\sqrt{2}},\,x^{-1/\sqrt{2}}y^{-1/\sqrt{2}}\\}$.
We will construct the restriction to $x=0$. The listed solutions imply that
the expected rank of the restriction is $1$, standing for the solution
$xy^{1/2}$, which is the only solution holomorphic at $x=0$. The $b$-function
for the weight $w=(1,0)$ is $(u-1)(u^{2}-1/2)$, implying that $u_{0}=1$. The
maximum $(-w,w)$-order of the $(-w,w)$-Gröbner basis of $\mathcal{I}$ is $0$,
hence $u_{1}=0$. Thus we put $\gamma=2$. We start with $k=1$ in Algorithm 4.4.
The submodule $\mathcal{J}$ is generated by

$\displaystyle(0,-1)=v_{1}(:f_{1}:|_{x=0}),$
$\displaystyle(2{y}\partial_{y}-1,0)=v_{1}(:\partial_{x}f_{1}:|_{x=0})=v_{1}(2y\partial_{x}\partial_{y}+2xy\partial_{x}^{2}\partial_{y}-\partial_{x}|_{x=0}),$
$\displaystyle(0,2{y}^{2}\partial_{y}^{2}+{y}\partial_{y}-1)=v_{1}(:f_{2}:|_{x=0}),$
$\displaystyle(2{y}^{2}\partial_{y}^{2}+{y}\partial_{y},0)=v_{1}(:\partial_{x}f_{2}:|_{x=0})$

The Gröbner basis of $\mathcal{J}$ with the POT order is

$\\{(0,-1),\,(2{y}\partial_{y}-1,0)\\},$

whose holonomic rank is $1$. The set of standard monomials of the Gröbner
basis in $\mathcal{D}^{2}$ is $S=\\{s_{1}:=(1,0)\in\mathcal{D}^{2}\\}$. Then,
we have the $1\times 1$ Pfaffian matrix $\left(\frac{1}{2y}\right)$ for $S$ as

$\partial_{y}s_{1}=(\partial_{y},0)\equiv\left(\frac{1}{2y},0\right)=\frac{1}{2y}s_{1}.$

$\blacksquare$

Let us now generalize to the case of several variables
$z=(z_{1},\ldots,z_{m})$. Fix an integer $m$ and a weight vector $w$
satisfying (92). We put

$c(k)=\\#E_{k}\>,\quad E_{k}=\bigl{\\{}\alpha\bigm{|}\alpha\cdot w\leq
k,\,\alpha_{1},\ldots,\alpha_{m^{\prime}}\in\mathbb{N}_{0},\,\alpha_{m^{\prime}+1}=\ldots=\alpha_{m}=0\bigr{\\}}\>.$
(98)

We regard
$\bigl{\\{}\partial_{1}^{\alpha_{1}}\ldots\partial_{m^{\prime}}^{\alpha_{m^{\prime}}}\bigm{|}\alpha\in
E_{k}\bigr{\\}}$ as the standard basis of ${\cal
D}_{Y^{\prime}}[z_{1},\ldots,z_{m^{\prime}}]^{c(k)}$. We define the map
$v_{k}$ from ${\cal D}_{Y}$ to ${\cal
D}_{Y^{\prime}}[z_{1},\ldots,z_{m^{\prime}}]^{c(k)}$ by

$v_{k}(\ell)=\sum_{\alpha\in
E_{k}}\ell_{\alpha}\partial_{1}^{\alpha_{1}}\ldots\partial_{m^{\prime}}^{\alpha_{m^{\prime}}}\,,$
(99)

where we decompose ${\cal D}\ni\ell$ as $\ell=\sum_{\alpha\in
E_{k}}\ell_{\alpha}\partial_{1}^{\alpha_{1}}\ldots\partial_{m^{\prime}}^{\alpha_{m^{\prime}}}$,
$\ell_{\alpha}\in{\cal D}_{Y^{\prime}}[z_{1},\ldots,z_{m^{\prime}}]$.
Algorithm 4.4 can be generalized using $v_{k}$. The map $v_{k}$ takes values
in a $c(k)$-dimensional vector space. We start the procedure from
$k\geq\max(u_{1},u_{0})$, where $u_{0}$ is the maximal non-negative integral
root of the $b$-function with respect to $w$, and $u_{1}$ is the maximum of
$\mathrm{ord}\,_{(-w,w)}$ for the $(-w,w)$-Gröbner basis of $\mathcal{I}$.
Note that $c(k)=k+1$ when $m=2,\,m^{\prime}=1$.

The case $m^{\prime}=m$ will be used to find a rational restriction by the
Macaulay matrix method, so the following algorithm assumes this equality.

######  Algorithm 4.7.

(Basis for the restriction to $z_{1}=\cdots=z_{m}=0$)
Input: Generators $\\{f_{1},\ldots,f_{\mu}\\}$ of a holonomic ideal
$\mathcal{I}$ in ${\cal D}_{Y}$. The vector space dimension $r$ of holomorphic
solutions at $z=(z_{1},\ldots,z_{m})=0$. A positive integer $\gamma$ such that
$\gamma\geq\max(u_{0},u_{1})$.
Output: A $\mathbb{C}$-basis of the restriction ${\cal N}_{0}$ to the point
$z=(z_{1},\ldots,z_{m})=0$.

1:$w=(1,\ldots,1)$

2:$k=\gamma$

3:repeat

4:
$\mathcal{J}=\mathbb{C}\cdot\Bigl{\\{}v_{k}\,\bigl{(}\,:\partial^{\alpha}f_{i}:{\big{|}_{z=0}}\bigr{)}\Bigm{|}\mathrm{ord}\,_{(-w,w)}\left(\prod_{j=1}^{m}\partial_{j}^{\alpha_{j}}f_{i}\right)\leq
k,\ \alpha\in\mathbb{N}_{0}^{m}\Bigr{\\}}\subseteq\mathbb{C}^{c(k)}$

5: $k:=k+1$

6:until
$\mathrm{dim}\bigl{(}\mathbb{C}^{c(\gamma)}/\mathcal{J}\cap\mathbb{C}^{c(\gamma)}\bigr{)}=r$

7:return A vector space basis of
$\mathbb{C}^{c(\gamma)}/\mathcal{J}\cap\mathbb{C}^{c(\gamma)}$.

Here we identify the vector space $\mathbb{C}^{c(k)}$ with the vector space
spanned by $\partial^{\alpha},\alpha\in E_{k}$. For example, when $m=2$,
$k=2$, and $\partial^{\alpha}$ is sorted as
$(\partial_{1}^{2},\partial_{1}\partial_{2},\partial_{2}^{2},\partial_{1},\partial_{2},1)$,
then $(0,0,0,0,0,1)$ stands for $1$, $(0,0,0,0,1,0)$ for $\partial_{2}$, and
$(1,0,0,0,0,0)$ for $\partial_{1}^{2}$. Note that this algorithm only utilizes
numerical linear algebra, so it is enough to compute a basis of the linear
space $\mathbb{C}^{c(\gamma)}/\mathcal{J}\cap\mathbb{C}^{c(\gamma)}$. The
numerical matrix representing $\mathcal{J}$ is large in general, so in our
implementations we perform the steps $6$ and $7$ over a finite field instead
of $\mathbb{C}$, which greatly speeds up the computation. Note also that we do
not know $\gamma$ beforehand without computing a Gröbner basis, for which
reasons we also choose $\gamma$ probabilistically in our implementation.

######  Example 4.8.

We denote $z_{1},z_{2}$ by $x,y$ respectively. The function
$(y^{2}-x^{3})^{1/2}$ spans the holomorphic solution space of the operators
$\\{2y\partial_{x}+3x^{2}\partial_{y},\,2x\partial_{x}+3y\partial_{y}-3\\}$
around $(x,y)=(1,2)$. Let us compute the restriction to this point. We make
the change of variables $x:=x+1$ and $y:=y+2$ and apply Algorithm 4.7 with
$w=(1,1)$ and $m=2$. We have $b(u)=u$, so $u_{0}=0$. The $(-w,w)$-initial
terms of the $(-w,w)$-Gröbner basis are

$\displaystyle\bigl{\\{}\partial_{x}+\ldots,\partial_{y}+\ldots,x\partial_{y}^{2}+\ldots,x^{2}\partial_{x}^{3}+\ldots\bigr{\\}}\>.$

The $(-w,w)$-orders all equal $1$. Hence, we can take $u_{1}=1$. We have
$E_{0}=\\{(0,0)\\}$ and $E_{1}=\\{(1,0),\,(0,1),\,(0,0)\\}$. Therefore, we can
set $\gamma=1$ and $c(\gamma)=2$. The input operators are

$\displaystyle f_{1}$
$\displaystyle=2(y+2)\partial_{x}+3(x+1)^{2}\partial_{y}$ (100) $\displaystyle
f_{2}$ $\displaystyle=2(x+1)\partial_{x}+3(y+2)\partial_{y}-3.$ (101)

Then the generators of $\mathcal{J}$ for $k=1$ are

$\displaystyle v_{k}(f_{1}|_{x=y=0})=(4,3,0)\quad,\quad
v_{k}(f_{2}|_{x=y=0})=(2,6,-3)\,,$ (102)

with the vector $v_{k}$ being indexed by $(\partial_{x},\partial_{y},1)$. The
row echelon form of
$\left[\begin{array}[]{ccc}4&3&{\color[rgb]{0.5,0.5,0.5}\cdot}\\\ 2&6&-3\\\
\end{array}\right]$ is
$\left[\begin{array}[]{ccc}-2&{\color[rgb]{0.5,0.5,0.5}\cdot}&-1\\\
{\color[rgb]{0.5,0.5,0.5}\cdot}&-3&2\\\ \end{array}\right],$ meaning that the
rank of $\mathbb{C}^{3}/\mathcal{J}\cap\mathbb{C}^{3}$ is $1$. Note that
$\gamma=1$ does not work even when we increase $k$ to larger values.
$\blacksquare$

###  4.2 Macaulay matrix method for restrictions

In this subsection, we consider the case when $Y^{\prime}\subset Y$ is an
irreducible, closed and smooth subvariety. Building on the work of
Chestnov:2022alh , the aim of this section is to give a Macaulay matrix method
for rational restrictions, as per Definition 2.11, when $Y^{\prime}$ is either
$z_{1}=\ldots=z_{m^{\prime}}=0$ or $Y^{\prime}$ is an irreducible
hypersurface. The rational restriction gives a description of holomorphic
solutions on $Y^{\prime}$.

Let ${\cal M}={\cal D}_{Y}/\mathcal{I}$ be a regular holonomic ${\cal
D}_{Y}$-module. The space $Y$ can be decomposed into algebraic varieties
$Y_{\alpha}$ such that $Y=\sqcup Y_{\alpha}$ and the space of holomorphic
solutions of ${\cal M}$ on $Y_{\alpha}$ is a locally constant sheaf. In other
words, all holomorphic solutions on $Y_{\alpha}$ are analytic continuations of
holomorphic solutions at a point in $Y_{\alpha}$ along paths in $Y_{\alpha}$.
In particular, the number of holomorphic solutions at a point $c\in
Y_{\alpha}$ only depends on $Y_{\alpha}$ and not $c$. This decomposition is
called a _stratification_. Stratifications were proved to exist in a broader
context in the work of M. Kashiwara kashiwara-1975 . For instance, a
stratification of the system in the Example 4.6 is

$\displaystyle\Bigl{(}\mathbb{C}^{2}\setminus
V(xy)\Bigr{)}\,\bigsqcup\,\Bigl{(}V(x)\setminus(0,0)\Bigr{)}\,\bigsqcup\,\Bigl{(}V(y)\setminus(0,0)\Bigr{)}\,\bigsqcup\,(0,0).$
(103)

Here, $V(f)$ is the variety defined by $f=0$. The number of holomorphic
solutions on these stratifications are respectively $4,1,0$ and $0$.

Consider the restriction $\mathcal{N}:=\iota^{*}{\cal M}$ and let $W\subset
Y^{\prime}$ be the maximum-dimensional stratum of $Y^{\prime}$. $W$ is known
to be a Zariski open set, and it does not intersect with the singular locus of
$\mathcal{N}$. We suppose that the origin $0$ belongs to $W$.

######  Lemma 4.9.

Let $S=\\{s_{i}\\}$ be a finite set consisting of monomials in $\partial$.
Assume that a set consisting of equivalence classes represented by elements of
$S$ is a $\mathbb{C}$-basis for a $\mathbb{C}$-vector space ${\cal
D}_{Y}/(\mathcal{I}+\sum_{i=1}^{m}z_{i}{\cal D}_{Y})$. Then the set of
equivalence classes in $\mathcal{N}|_{W}$ represented by elements of the form
$1\otimes s_{i}\in\mathcal{N}|_{W}$, for some $s_{i}\in S$, gives a free basis
of $\mathcal{N}|_{W}$ as an ${\cal O}_{Y^{\prime}}$-module.

It follows from this lemma that $S$ gives a basis for the rational restriction
of ${\cal M}$ to $Y^{\prime}$. The lemma is well-known folklore in the theory
of ${\cal D}$-modules and we give a constructive and elementary proof in the
appendices D.10, D.12.

Let $\\{f_{1},\ldots,f_{\mu}\\}$ be a set of generators for a regular
holonomic ideal $\mathcal{I}$ in $\mathcal{D}_{Y}$. We make a parallel change
of variables
$(z_{1},\ldots,z_{m})=(z^{\prime}_{1},\ldots,z^{\prime}_{m})+(c_{1},\ldots,c_{m})$
by choosing a numerical vector $(c_{1},\ldots,c_{m})$ in $W\subset Y^{\prime}$
such that we can apply Lemma 4.9. We dispose of the old coordinates $z_{i}$
and denote the new coordinates $z^{\prime}_{i}$ by $z_{i}$. A set of monomials
$S$ as in Lemma 4.9 can be obtained by Algorithm 4.7. Since we do not know the
stratum $W$, the chosen variable shift is probabilistic. If we unluckily
choose a point outside of $W$, the method fails. However, the relative measure
of $Y^{\prime}\setminus W$ is $0$, so the algorithm succeeds with probability
$1$. The following theorem easily follows from Lemma 4.9.

######  Theorem 4.10.

Let $S$ be as in Lemma 4.9, and regard it as a column vector. There exists an
$r\times\mu$ matrix $Q_{i}$ with entries in ${\cal
R}_{Y^{\prime}}[\partial_{1},\ldots,\partial_{m^{\prime}}]$ and an $r\times r$
matrix $P_{i}$ with entries in
$\mathbb{C}(z^{\prime}):=\mathbb{C}(z_{m^{\prime}+1},\ldots,z_{m})$ such that

$\partial_{i}S=P_{i}\cdot S\
+:Q_{i}\cdot[f_{1},\ldots,f_{\mu}]^{T}:\big{|}_{z_{1}=\ldots=z_{m^{\prime}}=0}$
(104)

holds in ${\cal R}_{Y^{\prime}}[\partial_{1},\ldots,\partial_{m^{\prime}}]$.

The expression (104) leads to the following Macaulay matrix method for finding
a Pfaffian system associated to the restriction to
$z_{1}=\ldots=z_{m^{\prime}}=0$.

######  Algorithm 4.11.

(Rationally restricted Pfaffian system via the Macaulay matrix)
Input: The basis $S$. Generators $\\{f_{1},\ldots,f_{\mu}\\}$ of a regular
holonomic ideal $\mathcal{I}$ in ${\cal D}_{m}$.
Output: Pfaffian matrices of the restriction ${\cal
R}_{Y^{\prime}}\otimes_{{\cal D}_{Y^{\prime}}}{\cal
D}_{Y}/(\mathcal{I}+z_{1}{\cal D}_{Y}+\ldots+z_{m^{\prime}}{\cal D}_{Y})$.

1:${\tt{RStd}}$=$S$.

2:Call find_macaulay_and_pfaffian($\\{f_{1},\ldots,f_{\mu}\\}$, ${\tt{RStd}}$,
$m^{\prime}$).

3:return The Pfaffian matrices $P_{i}$, $i=m^{\prime}+1,\ldots,m$.

Here find_macaulay_and_pfaffian($\\{f_{1},\ldots,f_{n}\\}$, ${\tt{RStd}}$,
$m^{\prime}$) is Algorithm 1 of Chestnov:2022alh with the following
modification in steps 2 and 6:

$M_{D}:=\>\left[\begin{array}[]{c|c}M_{{\tt{Ext}}}&M_{{\tt{RStd}}}\end{array}\right]\
\Big{|}_{z_{1}=\ldots=z_{m^{\prime}}=0}\>.$ (105)

The Macaulay matrix $M_{D}$ is constructed from the coefficients of certain
normally ordered differential operators. These operators come from acting on
the generators of $\mathcal{I}$ with a set of monomials
${{\tt{Mons}}}:=\\{\prod_{d_{1}+\cdots+d_{m}\leq
D,d_{i}\in\mathbb{Z}_{0}}\partial_{1}^{d_{1}}\dots\partial_{m}^{d_{m}}\\}\>$,
where the integer $D$ is called the degree of the Macaulay matrix. The columns
of $M_{D}$ are thus indexed by ${{\tt{Mons}}}$. Setting
${{\tt{Ext}}}:={{\tt{Mons}}}\setminus{{\tt{RStd}}}$, we obtain matrix blocks
$M_{{\tt{RStd}}}$ and $M_{{\tt{Ext}}}$ that are indexed by the monomials in
${{\tt{RStd}}}$ and ${{\tt{Ext}}}$ respectively. Note that ${{\tt{RStd}}}$ can
be a subset of a standard basis ${{\tt{Std}}}$ of
$\mathcal{R}_{m}/\mathcal{R}_{m}\mathcal{I}\>$.

With these modifications in comparison to Chestnov:2022alh , let us briefly
describe the Macaulay matrix method for finding Pfaffian systems. First,
define the matrices $C_{{\tt{Ext}}}$ and $C_{{\tt{RStd}}}$, with entries
consisting of $1$’s and $0$’s, by the expression

$\displaystyle\partial_{i}\,{{\tt{RStd}}}=:C_{{\tt{Ext}}}\cdot{{\tt{Ext}}}+C_{{\tt{RStd}}}\cdot{{\tt{RStd}}}\,.$
(106)

Here ${{\tt{RStd}}}$ and ${{\tt{Ext}}}$ are regarded as column vectors. The
Macaulay matrix method then instructs us to solve the linear equation

$C_{{\tt{Ext}}}-C\cdot M_{{\tt{Ext}}}|_{z_{1}=\ldots=z_{m^{\prime}}=0}=0$
(107)

for an unknown matrix $C$. The Pfaffian matrix in direction $\partial_{i}$ is
then given by

$\displaystyle P_{i}(z^{\prime})=C_{{\tt{RStd}}}-C\cdot
M_{{\tt{RStd}}}|_{z_{1}=\ldots=z_{m^{\prime}}=0}\,.$ (108)

See (Chestnov:2022alh, , Section 4) for additional details.

######  Remark 4.12.

Suppose that we know the holonomic rank $r$ of the restriction. One strategy
to find $S$ is to use the probabilistic method of Algorithm 4.7, as explained
after Lemma 4.9. A second strategy is to try all possible standard monomials
until the Macaulay matrix method succeeds.

######  Example 4.13.

We consider a left ideal generated by

$\displaystyle\partial_{x}(\theta_{x}+c_{1}-1)-(\theta_{x}+\theta_{y}+a)(\theta_{x}+b_{1})$
(109)
$\displaystyle\partial_{y}(\theta_{y}+c_{2}-1)-(\theta_{x}+\theta_{y}+a)(\theta_{y}+b_{2})\,,$
(110)

where $\theta_{x}=x\partial_{x}$ and $\theta_{y}=y\partial_{y}$. This system
annihilates the Appell $F_{2}$ function (see encyclopedia for a definition).
Let us try to find the restriction to $x=0$ by applying Algorithm 4.7 with
$\gamma=2,k=4$ and Algorithm 4.11 with $\gamma=1$ and $r=2$. We have
implemented the algorithms in Risa/Asir in the package mt_mm.rr url-asir . The
output ${\tt{RStd}}$ (a standard basis for the rational restriction) is
$[1,\partial_{y}]$ and the Pfaffian matrix P2 is constructed by the Macaulay
matrix method as follows.

import("mt_mm.rr")$Ideal = [(-x^2+x)*dx^2+(-y*x)*dx*dy+((-a-b1-1)*x+c1)*dx-b1*y*dy-b1*a, (-y^2+y)*dy^2+(-x*y)*dy*dx+((-a-b2-1)*y+c2)*dy-b2*x*dx-b2*a]$Xvars = [x,y]$//Rule gives a probabilistic determination of RStd (Std for the restriction)Rule=[[y,y+1/3],[a,1/2],[b1,1/3],[b2,1/5],[c1,1/7],[c2,1/11]]$Ideal_p = base_replace(Ideal,Rule);RStd=mt_mm.restriction_to_pt_(Ideal_p,Gamma=2,KK=4,[x,y] | p=10^8);RStd=reverse(map(dp_ptod,RStd[0],[dx,dy]));Id = map(dp_ptod,Ideal,poly_dvar(Xvars))$MData = mt_mm.find_macaulay(Id,RStd,Xvars | restriction_var=[x]);//For larger problems, use FiniteFlow instead of find_pfaffianP2 = mt_mm.find_pfaffian(MData,Xvars,2 | use_orig=1);The output P2 is

$\displaystyle P_{2}=\left[\begin{array}[]{cc}0&1\\\
\frac{-{b}_{2}{a}}{{y}({y}-1)}&\frac{-({a}+{b}_{2}+1){y}+{c}_{2}}{{y}({y}-1)}\\\
\end{array}\right]\,,$ (113)

which appears in the Pfaffian system for the restriction as
$\partial_{y}S=P_{2}\cdot S$, $S=[1,\partial_{y}]^{T}$.

Programs for larger problems are posted at dataAndProgramsOfThisPaper .
$\blacksquare$

The method presented above can also be applied to restrictions to irreducible
hypersurfaces. Suppose that $Y^{\prime}$ is the vanishing locus of an
irreducible polynomial $L\in\mathbb{C}[z]=\mathbb{C}[z_{1},\ldots,z_{m}]$ and
that $Y^{\prime}$ is non-singular. We then have the following theorem, which
is proved in Appendix D.11:

######  Theorem 4.14.

Let $S$ be a set as in Lemma 4.9 and regard it as a column vector. There
exists an $r\times\mu$ matrix $Q_{i}$ with entries in ${\cal D}_{Y}$, an
$r\times r$ matrix $P_{i}$ with entries in $\mathbb{C}[z]$ and a polynomial
$q_{i}\in\mathbb{C}[z]$ such that

$q_{i}\partial_{i}S=P_{i}\cdot S\
+:Q_{i}\cdot[f_{1},\ldots,f_{\mu}]^{T}:\quad\mathrm{mod}\,L$ (114)

holds in ${\cal D}_{Y}$. Here ${\cal D}_{Y}\ni\sum
c_{\alpha}(z)\partial^{\alpha}=0\ \mathrm{mod}\,L$ means that each
$c_{\alpha}(z)$ is divisible by $L$.

We call $P_{i}/q_{i}$ the Pfaffian matrix on the codimension-$1$ stratum $W$.

Let us now give a procedure for obtaining $P_{i}$ and $q_{i}$. Our column
vector $S$ will be denoted by ${\tt{RStd}}$. In this context, ${{\tt{Mons}}}$
consists of the monomials in $\partial$ appearing in
$:\partial^{\alpha}f_{j}:$ which construct the Macaulay matrix. Working modulo
$L$, the equation (107) is now modified to

$\displaystyle C_{{\tt{Ext}}}-C\cdot M_{{\tt{Ext}}}\equiv 0\quad\text{mod
}L\,,$ (115)

where the unknown matrix $C$ is solved for over the fraction field of the
quotient ring $\mathcal{O}_{Y^{\prime}}=\mathbb{C}[z]/L\mathbb{C}[z]$. In
other words, we find a $C$ with rational function entries and an
$|{{\tt{RStd}}}|\times|{{\tt{Ext}}}|$-dimensional matrix $H$ with polynomial
entries such that

$q_{i}(C_{{\tt{Ext}}}-C\cdot M_{{\tt{Ext}}})=HL$ (116)

holds, where $q_{i}$ is a polynomial which is relatively prime to $L$ that
cancels the denominators of the elements of $C$. Note that the entries of
$M_{{\tt{Ext}}}$ are polynomial in $z$. Then (116) can be solved by a syzygy
computation in the polynomial ring $\mathbb{C}[z]$. For example, when
$C_{{\tt{Ext}}}=[1,0]$,
$M_{{\tt{Ext}}}=\left[\begin{array}[]{cc}m_{11}&m_{12}\\\ m_{21}&m_{22}\\\
\end{array}\right]$, $H=[H_{1},H_{2}]$ and $C=[c_{1}/q_{i},c_{2}/q_{i}]$, we
may solve the syzygy equation

$\displaystyle q_{i}\left[\begin{array}[]{c}1\\\ 0\\\
\end{array}\right]-c_{1}\left[\begin{array}[]{c}m_{11}\\\ m_{12}\\\
\end{array}\right]-c_{2}\left[\begin{array}[]{c}m_{21}\\\ m_{22}\\\
\end{array}\right]-H_{1}\left[\begin{array}[]{c}L\\\ 0\\\
\end{array}\right]-H_{2}\left[\begin{array}[]{c}0\\\ L\\\
\end{array}\right]=\left[\begin{array}[]{c}0\\\ 0\\\ \end{array}\right]$ (129)

for the unknowns $H_{1},\,H_{2},\,c_{2},\,c_{2}$ and $q_{i}$. This can be
solved via a Gröbner basis computation adams . Finally, the matrix

$\displaystyle\frac{\widetilde{P}_{i}}{q_{i}}=C_{{\tt{RStd}}}-{C}\cdot
M_{{\tt{RStd}}}$ (130)

gives a Pfaffian matrix on the codimension-$1$ stratum $W$. Note that the
entries of $\widetilde{P}_{i}/q_{i}$ belong to the fraction field of the ring
$\mathcal{O}_{Y^{\prime}}$.

In Section 5.6, we use this procedure to find the restriction to the
hypersurface singularity $(x-y)^{2}-2(x+y)+1=0$ of the Appell $F_{4}$
hypergeometric system (see encyclopedia for a definition) and Horn’s
hypergeometric function $H_{i}$, $i=1,\dots,7$. As far as we are aware, the
results on $H_{i}$ were not known before.

##  5 Examples

We have explained general algorithms for restriction of Pfaffian systems in
Section 3, and restrictions for regular holonomic ${\cal D}$-modules in
Section 4. Now we apply these methods to Pfaffian systems associated to
Feynman integrals and GKZ-hypergeometric systems. Data and code for these
examples can be found in dataAndProgramsOfThisPaper .

When no confusion arises, $0$-entries in matrices and vectors are denoted by
${\color[rgb]{0.5,0.5,0.5}\cdot}$ in this section.

###  5.1 From GKZ to N-box

In this first example, we compute the restriction of a GKZ system at the level
of Pfaffian matrices, i.e. by the holomorphic restriction Algorithm 3.8.

####  5.1.1 Setup

Let $I_{\nu_{1},\ldots,\nu_{5}}$ denote the $2$-loop massless N-box integral
family presented in Example 2.2. We study this family as a restriction of a
GKZ system:

$\displaystyle A=\left[\begin{array}[]{ccccc ccccc}1&1&1&1&1&1&1&1&1&1\\\
1&1&1&{\color[rgb]{0.5,0.5,0.5}\cdot}&{\color[rgb]{0.5,0.5,0.5}\cdot}&{\color[rgb]{0.5,0.5,0.5}\cdot}&{\color[rgb]{0.5,0.5,0.5}\cdot}&{\color[rgb]{0.5,0.5,0.5}\cdot}&1&{\color[rgb]{0.5,0.5,0.5}\cdot}\\\
1&{\color[rgb]{0.5,0.5,0.5}\cdot}&{\color[rgb]{0.5,0.5,0.5}\cdot}&1&1&1&{\color[rgb]{0.5,0.5,0.5}\cdot}&{\color[rgb]{0.5,0.5,0.5}\cdot}&1&1\\\
{\color[rgb]{0.5,0.5,0.5}\cdot}&{\color[rgb]{0.5,0.5,0.5}\cdot}&{\color[rgb]{0.5,0.5,0.5}\cdot}&1&{\color[rgb]{0.5,0.5,0.5}\cdot}&{\color[rgb]{0.5,0.5,0.5}\cdot}&1&1&{\color[rgb]{0.5,0.5,0.5}\cdot}&1\\\
{\color[rgb]{0.5,0.5,0.5}\cdot}&1&{\color[rgb]{0.5,0.5,0.5}\cdot}&{\color[rgb]{0.5,0.5,0.5}\cdot}&1&{\color[rgb]{0.5,0.5,0.5}\cdot}&1&{\color[rgb]{0.5,0.5,0.5}\cdot}&1&{\color[rgb]{0.5,0.5,0.5}\cdot}\\\
{\color[rgb]{0.5,0.5,0.5}\cdot}&{\color[rgb]{0.5,0.5,0.5}\cdot}&1&{\color[rgb]{0.5,0.5,0.5}\cdot}&{\color[rgb]{0.5,0.5,0.5}\cdot}&1&{\color[rgb]{0.5,0.5,0.5}\cdot}&1&{\color[rgb]{0.5,0.5,0.5}\cdot}&1\end{array}\right]\quad\xrightarrow[i\,\neq\,10]{z_{i}=1}\quad\parbox[c]{0.0pt}{\includegraphics{figures/nbox}}$
(137)

While the Lee-Pomeransky polynomial for the GKZ system has generic monomial
coefficients $z_{1},\ldots,z_{10}$, to match the proper N-box Feynman integral
family we must to take the limits

$\displaystyle z_{1},\ldots,z_{9}\to 1\quad,\quad z_{10}\to\frac{t}{s}\,.$
(138)

As noted in Example 2.2, this restriction drops the rank from $9$ to $3$. The
goal of this example is to obtain the $3\times 3$ Pfaffian matrix
$Q_{10}(z_{10})$ by the holomorphic restriction procedure.

Before employing the restrictions at the level of Pfaffians, let us first use
the homogeneity property described in (Chestnov:2022alh, , Appendix A) to
rescale 6 ($=5+1$, where 5 is the number of integration variables) variables
to 1:

$\displaystyle z_{1},\,z_{2},\,z_{3},\,z_{4},\,z_{5},\,z_{9}\to 1\,.$ (139)

In the notation of (Chestnov:2022alh, , Appendix A), this corresponds to
choosing a _simplex_ $\sigma=[1,2,3,4,5,9]$. This rescaling is the maximal
restriction which does not change the rank of the GKZ system.

By the method of Hibi-Nishiyama-Takayama-2017 , we obtain $9$ standard
monomials for the rescaled GKZ system:

$\displaystyle{{\tt{Std}}}=\left[\partial_{8}^{2},\>\partial_{6}\partial_{10},\>\partial_{7}\partial_{10},\>\partial_{10}^{2},\>\partial_{6},\>\partial_{7},\>\partial_{8},\>\partial_{10},\>1\right]^{T}\,.$
(140)

The Macaulay matrix method of Chestnov:2022alh yields a Pfaffian system

$\displaystyle\partial_{i}{{\tt{Std}}}=P_{i}\cdot{{\tt{Std}}}\quad,\quad
i=6,7,8,10.$ (141)

We want to apply the restriction $z_{6},z_{7},z_{8}\to 1$ onto this system.

The $3$-dimensional integral basis $I$ for the restricted system is taken to
be

$\displaystyle\left[\begin{array}[]{c}I_{02202}\\\ I_{22020}\\\
I_{11111}\end{array}\right]=G_{0}\cdot I\,,$ (145)

where the mass dimension is factored out via $G_{0}$, a diagonal matrix
containing powers of $s$. By (Chestnov:2022alh, , Proposition A.1), we can
translate this integral basis into a $\mathcal{D}_{Y}$-module basis, finding

$\displaystyle
I_{02202}\quad\to\quad\frac{c_{1}}{\epsilon(\epsilon-1)}\partial_{10}^{2}$
(146) $\displaystyle
I_{22020}\quad\to\quad\frac{c_{2}}{\epsilon(\epsilon-1)}\Bigl{[}\epsilon(2+5\delta)(1+2\epsilon+5\epsilon\delta)+2z_{10}(1+2\epsilon+5\epsilon\delta)\partial_{10}+z_{10}^{2}\partial_{10}^{2}\Bigr{]}$
$\displaystyle
I_{11111}\quad\to\quad\frac{c_{3}}{\epsilon(\epsilon-1)}\Bigl{[}(1+\epsilon+2\delta\epsilon)\partial_{10}+z_{10}\partial_{10}^{2}-z_{7}\partial_{7}\partial_{10}+z_{6}\partial_{6}\partial_{10}\Bigr{]}\,,$

where we recall that $\epsilon$ and $\delta$ are analytic regulators stemming
from the integral representation (14). The prefactors

$\displaystyle
c_{1}=\tfrac{s^{-4-\epsilon-9\delta\epsilon}\Gamma(2-\epsilon)}{\Gamma(\delta\epsilon)^{6}\Gamma(-3\epsilon-9\delta\epsilon)\Gamma(2+\delta\epsilon)^{3}}\,,\>c_{2}=\tfrac{s^{-4-\epsilon-9\delta\epsilon}\Gamma(2-\epsilon)}{\Gamma(\delta\epsilon)^{6}\Gamma(-3\epsilon-9\delta\epsilon)\Gamma(2+\delta\epsilon)^{3}}\,,\>c_{3}=\tfrac{\epsilon^{2}(1+2\epsilon)s^{-4-\epsilon-9\delta\epsilon}\Gamma(2-\epsilon)}{\Gamma(\delta\epsilon)^{4}\Gamma(1-3\epsilon-9\delta\epsilon)\Gamma(1+\delta\epsilon)^{5}}$
(147)

come from the Lee-Pomeransky representation together with factors of $s$ from
$G_{0}$.

####  5.1.2 Normal form

We choose to do the restriction in the order131313We have checked that the
final result does not depend on the order of the individual limits. $z_{6}$,
$z_{7}$, $z_{8}$ . For the application of the restriction method, we must be
careful in checking that the Pfaffian system is in normal form (recall Section
3.2). We observe that the system is in normal form w.r.t. the variables
$z_{6}$ and $z_{7}$. However, the system is not logarithmic w.r.t. $z_{8}$:

$\displaystyle P_{8}\big{|}_{z_{6},z_{7}\to
1}=(z_{8}-1)^{-2}\>P_{8,-2}\big{|}_{z_{6},z_{7}\to
1}+(z_{8}-1)^{-1}\>P_{8,-1}\big{|}_{z_{6},z_{7}\to 1}+\ldots$ (148)

This 2nd order pole can be cured by Moser reduction, in this case a gauge
transformation by

$\displaystyle
G=\texttt{Diag[}\,(z_{8}-1)^{-1},1,\ldots,1\,\texttt{]}\quad,\quad
G[P_{i}]=G^{-1}(P_{i}\cdot G-\partial_{i}G)\,.$ (149)

We check that the transformed residue matrices $G[P_{i}]_{-1}$, for $i=6,7,8$,
all have non-resonant spectra. Hence, the Pfaffian system is now in normal
form w.r.t. all the variables we seek to restrict. To simplify the notation in
the following, let us

replace $P_{i}\to G_{1}[P_{i}]$ in this example

to denote the gauge transformed Pfaffian system.

####  5.1.3 Restriction

The restriction procedure implores us to build the matrix

$\displaystyle M=\left[\begin{array}[]{c}B\\\ \hline\cr R\end{array}\right]$
(152)

coming from (48). The block matrix $B$ translates between the unrestricted,
$9$-dimensional basis and the restricted, $3$-dimensional basis. This requires
us to expand the basis (146) in terms of the ${\tt{Std}}$ from (140)141414It
can happen that monomials in $\partial_{i}$ appear in the restricted basis
which are not part of ${\tt{Std}}$. This situation is treated in
(Chestnov:2022alh, , Section 3.2)., leading to the $3\times 9$ matrix

$\displaystyle B=\frac{1}{\epsilon(\epsilon-1)}\left[\begin{array}[]{ccccc
ccccc}{\color[rgb]{0.5,0.5,0.5}\cdot}&{\color[rgb]{0.5,0.5,0.5}\cdot}&{\color[rgb]{0.5,0.5,0.5}\cdot}&1&{\color[rgb]{0.5,0.5,0.5}\cdot}&{\color[rgb]{0.5,0.5,0.5}\cdot}&{\color[rgb]{0.5,0.5,0.5}\cdot}&{\color[rgb]{0.5,0.5,0.5}\cdot}&{\color[rgb]{0.5,0.5,0.5}\cdot}\\\\[4.0pt]
{\color[rgb]{0.5,0.5,0.5}\cdot}&{\color[rgb]{0.5,0.5,0.5}\cdot}&{\color[rgb]{0.5,0.5,0.5}\cdot}&z_{10}^{2}&{\color[rgb]{0.5,0.5,0.5}\cdot}&{\color[rgb]{0.5,0.5,0.5}\cdot}&{\color[rgb]{0.5,0.5,0.5}\cdot}&2z_{10}(1+2\epsilon+5\delta\epsilon)&\epsilon(2+5\delta)(1+2\epsilon+5\delta\epsilon)\\\\[4.0pt]
{\color[rgb]{0.5,0.5,0.5}\cdot}&z_{6}&-z_{7}&z_{10}&{\color[rgb]{0.5,0.5,0.5}\cdot}&{\color[rgb]{0.5,0.5,0.5}\cdot}&{\color[rgb]{0.5,0.5,0.5}\cdot}&1+\epsilon+2\delta\epsilon&{\color[rgb]{0.5,0.5,0.5}\cdot}\end{array}\right]\,,$
(156)

with columns labeled by ${\tt{Std}}$, and rows labeled by $I$. The block
matrix $R$ is built from the independent rows of the residue matrices:

$\displaystyle
R=\texttt{RowReduce\leavevmode\resizebox{}{9.95863pt}{[}}\,\left[\begin{array}[]{c}P_{6,-1}\\\
P_{7,-1}\\\
P_{8,-1}\end{array}\right]\,\texttt{\leavevmode\resizebox{}{9.95863pt}{]}}\ .$
(160)

Here we have stacked the matrices on top of each other, row reduced, and
finally deleted zero-rows.

Next we insert $M$ into (52):

$\displaystyle\Bigl{(}\partial_{10}M+M\cdot P_{10,0}\Bigr{)}\cdot
M^{-1}=\left[\begin{array}[]{cc}Q_{10}&\star\\\ \mathbb{0}_{\scriptscriptstyle
6\times 3}&\star\end{array}\right]\,,$ (163)

where $P_{10,0}$ is defined by the limit $P_{10}\big{|}_{z_{6},z_{7},z_{8}\to
0}\>$, which is a well-defined because we are in normal form. Sending the
analytic regular $\delta\to 0$ in $Q_{10}$, we finally obtain:

$\displaystyle
Q_{10}=\left[\begin{array}[]{ccc}-\frac{2(1+\epsilon)}{z_{10}}&{\color[rgb]{0.5,0.5,0.5}\cdot}&{\color[rgb]{0.5,0.5,0.5}\cdot}\\\
{\color[rgb]{0.5,0.5,0.5}\cdot}&{\color[rgb]{0.5,0.5,0.5}\cdot}&{\color[rgb]{0.5,0.5,0.5}\cdot}\\\
-\frac{z_{10}\epsilon}{6(1+z_{10})}&\frac{\epsilon}{6z_{10}(1+z_{10})}&-\frac{z_{10}+2\epsilon}{z_{10}(1+z_{10})}\end{array}\right]\,.$
(167)

This is the $3$-dimensional Pfaffian matrix for the restricted system. This
result agrees with an independent calculation performed with the IBP software
LiteRedLee:2012cn ; Lee:2013mka .

###  5.2 Logarithmic corrections to the one-loop Bhabha box integral

Let us now turn to non-GKZ example to illustrate Algorithm 3.9.

####  5.2.1 Setup

We would like to study logarithmic corrections in the small-mass expansion of
the following topology:

$\displaystyle\parbox[c]{0.0pt}{\includegraphics{figures/bhabha1}}\quad\xrightarrow{m\to
0}\quad\parbox[c]{0.0pt}{\includegraphics{figures/box}}$

The corresponding family of Feynman integrals in $d=4-2\epsilon$ dimensions is

$\displaystyle
I_{\nu_{1}\nu_{2}\nu_{3}\nu_{4}}(m)=\int\frac{\mathrm{d}^{d}k/i\pi^{d/2}}{\bigl{[}k^{2}-m^{2}\bigr{]}^{\nu_{1}}\bigl{[}(k+p_{1})^{2}\bigr{]}^{\nu_{2}}\bigl{[}(k+p_{12})^{2}-m^{2}\bigr{]}^{\nu_{3}}\bigl{[}(k+p_{123})^{2}\bigr{]}^{\nu_{4}}}\,,$
(168)

with $p_{K}:=\sum_{k\in K}p_{k}$ for $K\subset\\{1,2,3,4\\}$. The kinematic
variables are given by

$\displaystyle p_{1}^{2}=p_{2}^{2}=p_{3}^{3}=p_{4}^{2}=m^{2}\quad,\quad
s=p_{12}^{2}\quad,\quad t=p_{23}^{2}\,.$ (169)

This family of integrals contributes to Bhabha scattering at $1$-loop order.

After IBP reduction, we find $5$ master integrals, rescaled by a matrix
$G_{0}$ to become unitless:

$\displaystyle\left[\begin{array}[]{c}I_{0010}(m)\\\ I_{0101}(m)\\\
I_{1010}(m)\\\ I_{0111}(m)\\\ I_{1111}(m)\end{array}\right]=:G_{0}\cdot
I\quad,\quad
G_{0}=(-s)^{-\epsilon}\,\texttt{Diag\leavevmode\resizebox{}{9.95863pt}{[}}\,(-s),\,1,\,1,\,(-s)^{-1},\,(-s)^{-2}\,\texttt{\leavevmode\resizebox{}{9.95863pt}{]}}\,.$
(175)

The master integrals $I$ obey a Pfaffian system

$\displaystyle\partial_{i}\,I=P_{i}\cdot I\quad,\quad
z_{1}=\frac{m^{2}}{-s}\quad,\quad z_{2}=\frac{-t}{-s}$ (176)

in terms of two $5\times 5$ Pfaffian matrices $P_{i}=P_{i}(z_{1},z_{2})$.

Our goal is to find an approximation to the master integrals $I$ which holds
in the limit $m\to 0$, or equivalently

$\displaystyle z_{1}\to 0\,.$ (177)

The approximation is found by the restriction procedure laid out in Section
3.4. In particular, we will find and solve _simpler_ Pfaffian systems in
comparison to (176), in the sense that they will be of smaller rank and
contain one variable less.

####  5.2.2 Normal form

In the basis $I$, we find that $P_{1}$ and $P_{2}$ respectively have 2nd and
1st order poles at $z_{1}=0$. Moreover, the spectrum of the residue matrix of
$P_{1}$ is resonant. We therefore need to bring the Pfaffian system to normal
form before we can apply the restriction algorithm. We find that a single
gauge transformation

$\displaystyle I\,\to\,G_{1}\cdot I\quad,\quad
G_{1}[P_{i}]:=G_{1}^{-1}(P_{i}\cdot G_{1}-\partial_{i}G_{1})$ (178)

by a diagonal matrix

$\displaystyle G_{1}=\texttt{Diag[}\,z_{1},1,1,1,1\,\texttt{]}$ (179)

does the trick. To simplify the notation, we

$\displaystyle\text{replace $P_{i}\to G_{1}[P_{i}]$ in this example}\,.$ (180)

Now the following expansions in $z_{1}$ hold true:

$\displaystyle P_{1}$
$\displaystyle=\frac{1}{z_{1}}\>P_{1,-1}(z_{2})+O(z_{1}^{0})$ (181)
$\displaystyle P_{2}$ $\displaystyle=z_{1}^{0}\>P_{2,0}(z_{2})+O(z_{1})\,,$
(182)

with $P_{1,-1}$ having unique eigenvalues $\\{0,-\epsilon\\}$, which is a non-
resonant spectrum.

Jordan decomposition of the residue matrix $P_{1,-1}$ provides information on
how to subdivide the approximated solution to $I$. More precisely, given

$\displaystyle\texttt{JordanDecomposition[}\,P_{1,-1}\,\texttt{]}=\left[\begin{array}[]{ccccc}0&{\color[rgb]{0.5,0.5,0.5}\cdot}&{\color[rgb]{0.5,0.5,0.5}\cdot}&{\color[rgb]{0.5,0.5,0.5}\cdot}&{\color[rgb]{0.5,0.5,0.5}\cdot}\\\
{\color[rgb]{0.5,0.5,0.5}\cdot}&0&{\color[rgb]{0.5,0.5,0.5}\cdot}&{\color[rgb]{0.5,0.5,0.5}\cdot}&{\color[rgb]{0.5,0.5,0.5}\cdot}\\\
{\color[rgb]{0.5,0.5,0.5}\cdot}&{\color[rgb]{0.5,0.5,0.5}\cdot}&0&{\color[rgb]{0.5,0.5,0.5}\cdot}&{\color[rgb]{0.5,0.5,0.5}\cdot}\\\
{\color[rgb]{0.5,0.5,0.5}\cdot}&{\color[rgb]{0.5,0.5,0.5}\cdot}&{\color[rgb]{0.5,0.5,0.5}\cdot}&-\epsilon&1\\\
{\color[rgb]{0.5,0.5,0.5}\cdot}&{\color[rgb]{0.5,0.5,0.5}\cdot}&{\color[rgb]{0.5,0.5,0.5}\cdot}&{\color[rgb]{0.5,0.5,0.5}\cdot}&-\epsilon\end{array}\right]\,,$
(188)

we observe two blocks: a $3\times 3$ block associated to the eigenvalue
$\lambda_{1}=0$ and a $2\times 2$ block associated to the eigenvalue
$\lambda_{2}=-\epsilon$. As in (73), we then expect the approximated solution
to take the form

$\displaystyle I^{(\log)}(z_{1},z_{2})\quad\overset{z_{1}\to 0}{=}\quad
z_{1}^{\lambda_{1}}I^{(\lambda_{1})}(z_{2})\,+\,z_{1}^{\lambda_{2}}\left[I^{(\lambda_{2},0)}(z_{2})+I^{(\lambda_{2},1)}(z_{2})\log(z_{1})\right]\,.$
(189)

The vector $I^{(\lambda_{1})}$ will be found from the solution to a rank-$3$
Pfaffian system, and the vectors $I^{(\lambda_{2},0)},I^{(\lambda_{2},1)}$
from the solution to a rank-$2$ Pfaffian system. The logarithm in $z_{1}$
appears due to the $"1"$ in the superdiagonal of the $\lambda_{2}$-eigenvalue
block.

####  5.2.3 Restriction w.r.t. eigenvalue $\lambda_{1}=0$

Here we describe how to find the $5$-dimensional vector $I^{(\lambda_{1})}$ by
solving a $3\times 3$ subsystem in the variable $z_{2}$ only. Why is this
subsystem of rank $3$? Recalling (55), this question is answered by the
following constraints which hold in the limit $z_{1}\to 0$:

$\displaystyle\texttt{RowReduce[}\,P_{2,-1}\,\texttt{]}\cdot
I^{(\lambda_{1})}$ $\displaystyle:=R^{(\lambda_{1})}\cdot I^{(\lambda_{1})}$
(190)
$\displaystyle=\left[\begin{array}[]{ccccc}1&{\color[rgb]{0.5,0.5,0.5}\cdot}&{\color[rgb]{0.5,0.5,0.5}\cdot}&{\color[rgb]{0.5,0.5,0.5}\cdot}&{\color[rgb]{0.5,0.5,0.5}\cdot}\\\
{\color[rgb]{0.5,0.5,0.5}\cdot}&1&{\color[rgb]{0.5,0.5,0.5}\cdot}&\frac{z_{2}\epsilon}{1-2\epsilon}&{\color[rgb]{0.5,0.5,0.5}\cdot}\end{array}\right]\,\cdot\,I^{(\lambda_{1})}$
(193) $\displaystyle=0\,.$ (194)

We may regard $I^{(\lambda_{1})}$ as the piece of the full solution vector $I$
which is holomorphic in the massless limit, wherefore we expect every entry of
$I^{(\lambda_{1})}$ to be proportional to $I_{i}(m=0)$, for some
$i=1,\ldots,5$, with the massless limit being taken at the _integrand_
level151515 This corresponds to the so-called “hard region” in the expansion
by regions method Beneke:1997zp . . The first row of (193) thus states that
the massless tadpole $I_{0010}(0)$ vanishes, and the 2nd row yields an IBP
relation between the massless $t$-channel bubble $I_{0101}(0)$ and the
massless triangle $I_{0111}(0)$. These two constraints drop the rank from $5$
to $3$. Summing up, $I^{(\lambda_{1})}$ is of the form

$\displaystyle
I^{(\lambda_{1})}=G_{0}\cdot\left[\begin{array}[]{c}{\color[rgb]{0.5,0.5,0.5}\cdot}\\\
I_{0101}(0)\\\ I_{1010}(0)\\\ \frac{2\epsilon-1}{z_{2}\epsilon}I_{0101}(0)\\\
I_{1111}(0)\end{array}\right].$ (200)

Next we find a 3-dimensional basis for the vector space associated to the
eigenvalue $\lambda_{1}=0$. One possible basis is

$\displaystyle\texttt{NullSpace[}\,P_{1,-1}\,\texttt{]}=\left[\begin{array}[]{ccccc}{\color[rgb]{0.5,0.5,0.5}\cdot}&\frac{z_{2}\epsilon}{2\epsilon-1}&{\color[rgb]{0.5,0.5,0.5}\cdot}&1&{\color[rgb]{0.5,0.5,0.5}\cdot}\\\
{\color[rgb]{0.5,0.5,0.5}\cdot}&{\color[rgb]{0.5,0.5,0.5}\cdot}&1&{\color[rgb]{0.5,0.5,0.5}\cdot}&{\color[rgb]{0.5,0.5,0.5}\cdot}\\\
{\color[rgb]{0.5,0.5,0.5}\cdot}&{\color[rgb]{0.5,0.5,0.5}\cdot}&{\color[rgb]{0.5,0.5,0.5}\cdot}&{\color[rgb]{0.5,0.5,0.5}\cdot}&1\end{array}\right]\,,$
(204)

where the rows of this matrix, viewed as vectors, span the nullspace of
$P_{1,-1}$. The 1st row suggests a linear combination between $I_{0101}(0)$
and $I_{0111}(0)$ as a basis element. We could use this as it is, but by
virtue of (193) it will be simpler to rewrite $I_{0111}(0)$ in terms of
$I_{0101}(0)$ up to a non-zero prefactor, which we are free to omit in our
choice of basis. The 2nd and 3rd rows of (204) respectively instruct us to
pick the $s$-channel bubble $I_{1010}(0)$ and the box $I_{1111}(0)$ as the
last two basis elements. Summing up, we have the following basis for the
restricted Pfaffian system associated to the eigenvalue $\lambda_{1}$:

$\displaystyle
J^{(\lambda_{1})}:=(-s)^{\epsilon}\left[\begin{array}[]{c}I_{0101}(0)\\\
I_{1010}(0)\\\ (-s)^{2}I_{1111}(0)\end{array}\right]\,,$ (208)

with prefactors of $s$ stemming from $G_{0}$ in (175). The $3\times 5$ matrix
relating the bases $J^{(\lambda_{1})}$ and $I^{(\lambda_{1})}$ is thus

$\displaystyle
B^{(\lambda_{1})}=\left[\begin{array}[]{ccccc}{\color[rgb]{0.5,0.5,0.5}\cdot}&1&{\color[rgb]{0.5,0.5,0.5}\cdot}&{\color[rgb]{0.5,0.5,0.5}\cdot}&{\color[rgb]{0.5,0.5,0.5}\cdot}\\\
{\color[rgb]{0.5,0.5,0.5}\cdot}&{\color[rgb]{0.5,0.5,0.5}\cdot}&1&{\color[rgb]{0.5,0.5,0.5}\cdot}&{\color[rgb]{0.5,0.5,0.5}\cdot}\\\
{\color[rgb]{0.5,0.5,0.5}\cdot}&{\color[rgb]{0.5,0.5,0.5}\cdot}&{\color[rgb]{0.5,0.5,0.5}\cdot}&{\color[rgb]{0.5,0.5,0.5}\cdot}&1\end{array}\right]\quad,\quad
J^{(\lambda_{1})}=B^{(\lambda_{1})}\cdot I^{(\lambda_{1})}\,.$ (212)

Using (48), we proceed to build the invertible matrix

$\displaystyle M^{(\lambda_{1})}=\left[\begin{array}[]{c}B^{(\lambda_{1})}\\\
\hline\cr R^{(\lambda_{1})}\end{array}\right]\,,$ (215)

which has the property

$\displaystyle\left[\begin{array}[]{c}J^{(\lambda_{1})}\\\
{\color[rgb]{0.5,0.5,0.5}\cdot}\\\
{\color[rgb]{0.5,0.5,0.5}\cdot}\end{array}\right]=M^{(\lambda_{1})}\cdot
I^{(\lambda_{1})}\,.$ (219)

The $3\times 3$ Pfaffian matrix $Q_{2}^{(\lambda_{1})}$ associated to the
vector $J^{(\lambda_{1})}$ is now found via (52):

$\displaystyle\Bigl{(}\partial_{2}M^{(\lambda_{1})}+M^{(\lambda_{1})}\cdot
P_{2,0}\Bigr{)}\cdot(M^{(\lambda_{1})})^{-1}=\left[\begin{array}[]{cc}Q_{2}^{(\lambda_{1})}&\star\\\
\mathbb{0}_{\scriptscriptstyle 2\times 3}&\star\end{array}\right]\,,$ (222)

with $P_{2,0}$ defined in (182). Explicitly, the Pfaffian system restricted to
$z_{1}=0$ w.r.t. the eigenvalue $\lambda_{1}=0$ is

$\displaystyle\partial_{2}J^{(\lambda_{1})}=Q_{2}^{(\lambda_{1})}\cdot
J^{(\lambda_{1})}\quad,\quad
Q_{2}^{(\lambda_{1})}=\left[\begin{array}[]{ccc}-\frac{\epsilon}{z_{2}}&{\color[rgb]{0.5,0.5,0.5}\cdot}&{\color[rgb]{0.5,0.5,0.5}\cdot}\\\
{\color[rgb]{0.5,0.5,0.5}\cdot}&{\color[rgb]{0.5,0.5,0.5}\cdot}&{\color[rgb]{0.5,0.5,0.5}\cdot}\\\
\frac{2(1-2\epsilon)}{z_{2}^{2}(z_{2}+1)}&\frac{2(2\epsilon-1)}{z_{2}(z_{2}+1)}&\frac{-z_{2}-\epsilon-1}{z_{2}(z_{2}+1)}\end{array}\right]\,.$
(226)

This agrees with the system one would find by performing IBP reduction
directly on the massless integrals $J^{(\lambda_{1})}$, as can be checked with
LiteRed. It is standard to solve such a system by passing to a canonical basis
Henn:2013pwa (see Henn:2014qga for this specific example).

####  5.2.4 Restriction w.r.t. eigenvalue $\lambda_{2}=-\epsilon$

Next we determine the $5$-dimensional vectors
$I^{(\lambda_{2},0)},I^{(\lambda_{2},1)}$ by solving a rank-$2$ Pfaffian
system. The rank is $2$ because of (76), which imposes the following
constraint on the solution vector $I^{(\lambda_{2},0)}$:

$\displaystyle\texttt{RowReduce\leavevmode\resizebox{}{9.95863pt}{[}}\,(P_{1,-1}-\lambda_{2})^{2}\,\texttt{\leavevmode\resizebox{}{9.95863pt}{]}}\cdot
I^{(\lambda_{2},0)}$ $\displaystyle:=R^{(\lambda_{2})}\cdot
I^{(\lambda_{2},0)}$ (227)
$\displaystyle=\left[\begin{array}[]{ccccc}1&{\color[rgb]{0.5,0.5,0.5}\cdot}&{\color[rgb]{0.5,0.5,0.5}\cdot}&\frac{z_{2}\epsilon}{1-\epsilon}&\frac{z_{2}\epsilon}{2(1-\epsilon)}\\\
{\color[rgb]{0.5,0.5,0.5}\cdot}&1&{\color[rgb]{0.5,0.5,0.5}\cdot}&{\color[rgb]{0.5,0.5,0.5}\cdot}&{\color[rgb]{0.5,0.5,0.5}\cdot}\\\
{\color[rgb]{0.5,0.5,0.5}\cdot}&{\color[rgb]{0.5,0.5,0.5}\cdot}&1&{\color[rgb]{0.5,0.5,0.5}\cdot}&{\color[rgb]{0.5,0.5,0.5}\cdot}\end{array}\right]\cdot
I^{(\lambda_{2},0)}$ (231) $\displaystyle=0\,.$ (232)

Since there are $3$ relations imposed on the entries of $I^{(\lambda_{2},0)}$,
the rank drops from $5$ to $2$. Let us further note that, according to (75),
we obtain $I^{(\lambda_{2},1)}$ for free once we know $I^{(\lambda_{2},0)}$:

$\displaystyle(P_{1,-1}-\lambda_{2})\cdot
I^{(\lambda_{2},0)}=I^{(\lambda_{2},1)}\,.$ (233)

Let

$\displaystyle
J^{(\lambda_{2})}=\left[\begin{array}[]{c}J_{1}^{(\lambda_{2})}\\\
J_{2}^{(\lambda_{2})}\end{array}\right]\,$ (236)

denote the basis of the Pfaffian system associated to the eigenvalue
$\lambda_{2}$. Unlike the zero eigenvalue case of Section 5.2.3, here we no
longer have a natural choice for the basis matrix $B^{(\lambda_{2})}$ that
relates $J^{(\lambda_{2})}$ with $I^{(\lambda_{2})}$. Luckily, when buidling
the transformation matrix $M^{(\lambda_{2})}$ from (48), we only need to
ensure that the our choice of $B^{(\lambda_{2})}$ together with the
$R^{(\lambda_{2})}$ of (232) produce a full rank square matrix, so let us
take161616 See also the procedure described in Appendix C.2.2 that gives an
alternative choice of the basis matrix.

$\displaystyle
B^{(\lambda_{2})}:=\left[\begin{array}[]{ccccc}{\color[rgb]{0.5,0.5,0.5}\cdot}&{\color[rgb]{0.5,0.5,0.5}\cdot}&{\color[rgb]{0.5,0.5,0.5}\cdot}&\frac{z_{2}\epsilon}{\epsilon-1}&\frac{z_{2}\epsilon}{2(\epsilon-1)}\\\
{\color[rgb]{0.5,0.5,0.5}\cdot}&{\color[rgb]{0.5,0.5,0.5}\cdot}&{\color[rgb]{0.5,0.5,0.5}\cdot}&1&{\color[rgb]{0.5,0.5,0.5}\cdot}\\\
\end{array}\right]\>.$ (239)

Using (48) to write

$\displaystyle M^{(\lambda_{2})}=\left[\begin{array}[]{c}B^{(\lambda_{2})}\\\
\hline\cr R^{(\lambda_{2})}\end{array}\right]\implies
I^{(\lambda_{2},0)}=\bigl{(}M^{(\lambda_{2})}\bigr{)}^{-1}\cdot\left[\begin{array}[]{c}J^{(\lambda_{2})}\\\\[-5.0pt]
{\color[rgb]{0.5,0.5,0.5}\cdot}\\\\[-5.0pt]
{\color[rgb]{0.5,0.5,0.5}\cdot}\\\\[-5.0pt]
{\color[rgb]{0.5,0.5,0.5}\cdot}\end{array}\right]=\left[\begin{array}[]{c}J_{1}^{(\lambda_{2})}\\\
{\color[rgb]{0.5,0.5,0.5}\cdot}\\\ {\color[rgb]{0.5,0.5,0.5}\cdot}\\\
J_{2}^{(\lambda_{2})}\\\
\frac{2(\epsilon-1)}{z_{2}\epsilon}J_{2}^{(\lambda_{2})}-2J_{2}^{(\lambda_{2})}\end{array}\right]\,,$
(251)

we then insert $M^{(\lambda_{2})}$ into (52) to obtain the $2\times 2$
Pfaffian matrix associated to $J^{(\lambda_{2})}$:

$\displaystyle\Bigl{(}\partial_{2}M^{(\lambda_{2})}+M^{(\lambda_{2})}\cdot
P_{2,0}\Bigr{)}\cdot\bigl{(}M^{(\lambda_{2})}\bigr{)}^{-1}=\left[\begin{array}[]{cc}Q_{2}^{(\lambda_{2})}&\star\\\
\mathbb{0}_{\scriptscriptstyle 3\times 2}&\star\end{array}\right]\,.$ (254)

Explicitly, the Pfaffian system restricted to $z_{1}=0$ w.r.t. the eigenvalue