the singular locus, if not transversally.
The first important request is that any chain has to meet $X_{2m-2}$
transversally. For algebraic varieties of complex dimension 1, that’s all. For
example, in the case of our glued würstel this just implies that the loop
around the hole is forbidden. Thus, $H_{1}(Z)=0$ and the perverse homology of
the singular surface is the same as for the two dimensional sphere. Indeed,
the singularity in this case is non normal, in the sense that if we take a
small neighbourhood of the singular point $p$ and keep the point out, the
neighbourhood separates in two disconnected components. We can normalize the
surface by adding the laking point separately to each one of the two
disconnected parts. In other words we unglue the würstel, which becomes an
usual one, with the topology of $S^{2}$. These kinds of singularity are thus
simply solved by normalization, and for the case of complex dimension two
things remain relatively simple.
However, for higher dimensional $X$, one has to consider also intersection
with higher codimensional singularities. Here is where one introduces a
perversity, [127, 128, 129, 130, 131, 132, 133, 134, 135]:

######  Definition 1.

A perversity is a function

$\displaystyle p:\mathbb{Z}_{\geq 2}\longrightarrow\mathbb{N},$ (233)

satisfying the conditions

$\displaystyle p(2)=0,\qquad p(j)\leq p(j+1)\leq p(j)+1,\quad j\geq 2.$ (234)

In particular, $p_{t}(j)=j-2$ is called the top perversity, while $p_{0}(j)=0$
is the zero perversity. Also, $p_{lm}(j)=\lfloor(j-2)/2\rfloor$, $j>2$, is
called the lower-middle perversity, and $p_{um}=p_{t}-p_{lm}$ is the upper-
middle perversity.
Given a perversity $p$, an allowed $k$-chain $\sigma$ is a locally finite
chain whose support $|\sigma|$ satisfies

$\displaystyle{\rm dim}(|\sigma|\cap X_{2m-j})$ $\displaystyle\geq k-j+p(j),$
(235) $\displaystyle{\rm dim}(|\partial\sigma|\cap X_{2m-j})$
$\displaystyle\geq k-j+p(j)-1,$ (236)

for all $j\geq 2$. The so generated homology groups are called the BM
intersection homology groups with perversity $p$ of $X$,
$I^{BM}H_{k}^{(p)}(X,\mathbb{Z})$. If in place of locally finite chains one
starts with finite chains, then one obtains the homology groups with
perversity $p$, $IH_{k}^{(p)}(X,\mathbb{Z})$.
With these notions, one has the following version of the Poincaré duality
([121], Th.2.6.1):

######  Theorem 6 (Poincaré duality).

If $X$ is an algebraic manifold of complex dimension $m$, whose singularities
define a filtration as above, and $p$ and $\bar{p}$ are two complementary
perversities, which means $p+\bar{p}=p_{t}$, then there is a a non-degenerate
bilinear pairing

$\displaystyle\frown:IH_{k}^{(p)}(X,\mathbb{Z})\times
I^{BM}H_{2m-k}^{(\bar{p})}(X,\mathbb{Z})\longrightarrow\mathbb{Q}.$ (237)

This theorem provides the necessary definition of ($\mathbb{Q}$-valued)
intersection number. A construction that proves homological invariance and
independence of choices, as well as calculability, passes through sheaf theory
and its generalizations. Analyzing such a formulation goes too far beyond the
aim of our discussion. However, we want just to add some remarks, for who has
at least a basic knowledge on sheaf theory (a nice introduction is in [121],
Chp. 4). In the sheaf theoretical formulation, a particular role is played by
Local Systems, which are locally constant sheaves, that is, any sheaf
$\mathcal{L}$ over $X$, such that its local restriction to a small
neighbourhood $U_{p}$ of any given $p\in X$, $\mathcal{L}_{p}\simeq M_{p}$, a
given $A$-module. Thus, it comes out to be necessary to work with twisted
homologies, which are homologies taking value in a Local System. This leads to
work with complexes of sheaves. A main difficulty in doing these, is that in
such framework often diagrams commute only up to homotopies and not
identically. To overcome this problem, one introduces the category $K(X)$,
that is essentially the same of the one of complex of sheaves, up to
identifications by homotopy. The problem is that one of the main computational
instruments, t.i. short exact sequences, are lost in passing from the category
$C(X)$ of complexes of sheaves to $K(X)$. A remedy consists in finding a way
to map short exact sequences in $C(X)$ into distinguished triangles, from
which it is also possible to construct long exact sequences. This leads to the
language of derived categories ([121], Chp. 4). This way one arrives to
construct the intersection homology groups $IH^{k}(X)$.
In place of embarking into the formidably difficult program of analyzing such
constructions further, we pass to illustrate a possible strategy to be applied
to the more specific situation of computing integrals of the form (2),
suggested by M. Kontsevich in a private communication to P. Mastrolia, T.
Damour and S.L. Cacciatori, at IHES on January 2020.

###  6.4 From Feynman integrals to intersection theory.

Let us consider (2). In general, we assume that $\nu_{j}=0$ for sake of
simplicity, and $\gamma\in\mathbb{C}\backslash\mathbb{Z}$.101010Indeed
$\gamma=N/2+\epsilon$ because of dimensional regularization. Consider the map

$\displaystyle
B:\mathbb{C}_{*}^{M}\longrightarrow\mathbb{C}\supset\mathbb{C}_{*}.$ (238)

This map has critical regions with critical values $t_{\alpha}\in\mathbb{C}$,
$\alpha=1,\ldots,n$. For each $t$, we can consider the algebraic variety

$\displaystyle X_{t}=B^{-1}(t)\cap\mathbb{C}_{*}^{M},$ (239)

with
$t\in\tilde{\mathbb{C}}\equiv\mathbb{C}^{*}-\\{t_{\alpha}\\}_{\alpha=1}^{n}$.
For these $t$, $X_{t}$ is smooth and admits a compactification $\bar{X}_{t}$,
with $\bar{X}_{t}$ projective, $X_{t}\subset\bar{X}_{t}$ open embedding, and
$\bar{X}_{t}\backslash X_{t}$ is a normal crossing divisor, which means that
at any point of $X_{t}$ it is described by local equations
$\prod_{j=1}^{r}z_{j}$. Thus, starting from $H^{M-1}(X_{t},\mathbb{Z})$, we
get a (Deligne) weight filtration, starting from $\bar{X}_{t}$ (with
$H^{M-1}(\bar{X}_{t})$ of lowest weight $0$). When $t$ moves in
$\tilde{\mathbb{C}}$, then we get a variation of mixed Hodge structures on
$H^{M-1}(X_{t},\mathbb{Z})$.
Now, these variations of mixed Hodge structures define a perverse sheaf over
$\tilde{\mathbb{C}}$. Luckily, on complex curves they can be understood quite
easily, we refer to [136] for a nice presentation. A perverse sheaf is not
exactly a sheaf, but has properties similar to a sheaf, as it can be glued by
local data. Also, away from a singular point, it is effectively a sheaf, more
precisely a Local System, so on $x$ non singular, we can think the relative
stalks as a finitely generated module $V$. Around a singular point, one has to
add a monodromy rule around the singular point $t_{\alpha}$:

$\displaystyle T_{\alpha}:V\rightarrow V,\qquad\alpha=1,\ldots,n.$ (240)

This is the effect of our variation of mixed Hodge monodromy. Let
$V^{T_{\alpha}}\subseteq V$ the submodule of invariant elements
(eigenelements). The exact sequence $V^{T_{\alpha}}\hookrightarrow
V\rightarrow V_{T_{\alpha}}$ defines the coinvariant module $V_{T_{\alpha}}$.
Since the monodromy action depends only on the first homotopy group, we can
see monodromies as representing the action of the first homotopy group
$\pi_{1}(\tilde{\mathbb{C}})$. On $V_{T_{\alpha}}$, we get an irreducible
representation of $\pi_{1}(\mathbb{C}^{*}-\\{t_{\alpha}\\}_{\alpha=1}^{n})$ on
a variation of a pure Hodge structure. Thus, it is endowed with a polarization
via the Hodge-Riemann bilinear relations. This polarization is symmetric or
skew-symmetric, according to the parity of $M-1$, so we have representations

$\displaystyle\pi_{1}(\tilde{\mathbb{C}})\longrightarrow G,$ (241)

with $G$ $\mathbb{Z}$-orthogonal or $\mathbb{Z}$-symplectic, according to the
given parity. Any such irreps determines univocally a local system. Notice
that from the Primitive Lefschetz decomposition one can read out the precise
structure of the intersection product induced by the polarization.
At this point, one can also determine a basis for the homology underlying the
original integral, using this perverse extension. We can get a twisted sheaf
over $C_{*}$, given by the Local System $\mathcal{L}$ twisted by the perverse
monodromies, associated to $t^{\gamma}$. The original integral can be thought
to be defined on a middle dimensional cycles in a complex $m$-dimensional
variety $Y$. The cohomology $H^{1}(\mathbb{C}_{*},\mathcal{L})$ determines a
basis of real $m$-dimensional Lefschetz thimbles in $Y$, Fig. 22, which indeed
represent the perverse cohomology structure.

$\boldsymbol{B}$$\boldsymbol{t_{1}}$$\boldsymbol{t_{2}}$$\boldsymbol{c^{1}_{t}}$$\boldsymbol{c^{2}_{t}}$$\boldsymbol{T}$
Figure 22: Thimbles drawn above the plane of $t=-\gamma\log B$.

On $\mathbb{C}$, each critical value $t_{\alpha}$ of $B$ determines a phase of
$t_{\alpha}^{\gamma}$. Draw $n$ lines with such constant phases, passing
through the $n$ critical values (so $\gamma\log B$ has constant imaginary
part) starting from the critical values of $B$ to $\infty$ or to zero,
according Re$(\gamma)<0$, or Re$(\gamma>0)$. To each line it corresponds a
family of $(m-1)$-dimensional cycles $c^{\alpha}_{t}$, collapsing to a point
in $t=t_{\alpha}$ (a collapsing cycle), and going towards a region where
$B^{-\gamma}$ assumes larger and larger values. Such cycles are determined by
the coinvariant elements in a way similar to the construction we gave in
section 5.4. Fix a large real value $T\gg{\rm Re}(-\gamma\log t_{\alpha})$,
for all $\alpha$; then $-\gamma\log t>T$ corresponds to the region $Z\subset
Y$ where $B^{-\gamma}|_{Z}>T$. The unions

$\displaystyle\tau^{\alpha}=\bigcup_{t|{\scriptsize\begin{pmatrix}{\rm
Im}(-\gamma\log\frac{t}{t_{\alpha}})=0,\\\ {\rm
Re}(-\gamma\log\frac{t}{t_{\alpha}})>0\end{pmatrix}}}c^{\alpha}_{t},$ (242)

are called Lefschetz thimbles, and result to be a basis for the middle
homology of $Y$ relative to $Z$. We remark that such constructions are quite
familiar to physicists even if in a different form. Let us define the
potential function

$\displaystyle W=-\hbar\gamma\log B$ (243)

so that the integrand becomes

$\displaystyle e^{\frac{W}{\hbar}}.$ (244)

One can evaluate the integral for very small $\hbar$ by using the stationary
phase method. So, such integral is dominated by the stationary points, where

$\displaystyle dW=0=\frac{dB}{B}.$ (245)

These are the critical points of the polynomial. The valuation of the integral
is then dominated by the paths that make the integrand as less oscillating as
possible. These are the union of paths such that the phase of the integrand is
constant, and that goes to infinity with Re$(W)>0$, so that the integrand goes
to zero. This means ${\rm Re}(-\gamma\log\frac{t}{t_{\alpha}})>0$, and we get
the thimbles.

Here, we end up our mathematical excursus and go back to the practical
constructions of Cho, Matsumoto and others.

##  7 Computing Intersection Numbers: state of art and open problems

In this section we mainly address the practical computation of the
coefficients of the Master Integrals, the so-called intersection numbers.
Intersection numbers between two $n$-forms $\bra{\phi_{L}}$ and
$\ket{\phi_{R}}$, which have been introduced abstractly in Section 6, are
defined by Cho and Matsumoto [74] as

$\bra{\phi_{L}}\ket{\phi_{R}}_{\omega}:=\frac{1}{(2\pi
i)^{n}}\int_{X}\iota_{\omega}(\phi_{L})\wedge\phi_{R}\,,$ (246)

where $X$ is the whole complex space $\mathbb{C}^{n}$ deprived of the
hyperplanes corresponding to the poles of $\omega$ and $\iota_{\omega}$ is a
map which sends $\phi_{L}$ to an equivalent form with compact support. In our
particular case, $\bra{\phi_{L}}$ and $\ket{\phi_{R}}$ are twisted cocycles,
which implies that the intersection number in Eq. (246) is not an integer in
general [137].
Notice that if we omitted $\iota_{\omega}$ in Eq. (246), then the intersection
number would vanish, as both $\phi_{L}$ and $\phi_{R}$ are holomorphic in the
domain of integration $X$. To understand why this is true, we focus on a 1D
description (there is only a single variable $z$) in order to have a simple
notation, but there is no difference in considering a multidimensional
example. We consider $z$ to be complex, hence $X$ is $\mathbb{C}$ without some
points corresponding to poles. A certain function $f(x,y)$ can always be split
into its real and imaginary part: $f(x,y)=u+iv$. Introducing a change of
variables

$\begin{cases}z=x+iy\\\ \bar{z}=x-iy\end{cases}$ (247)

allows to obtain a certain $\tilde{f}(z,\bar{z})$. If $f$ is holomorphic, in
order to satisfy the Cauchy-Riemann conditions, after the change of variables
$\tilde{f}$ depends only on $z$: $\tilde{f}(z,\bar{z})=f(z)$. In general a
form $\phi$ can be decomposed as

$\phi=\hat{\phi}_{1}dz+\hat{\phi}_{2}d\bar{z}\,.$ (248)

When we wedge two forms as in Eq. (248), only the mixed terms $dz\wedge
d\bar{z}$ survive; but if the forms involved are holomorphic, the wedge
product is null.

Strictly focusing on the twisted cohomology framework, we present in this
section a summary on what has been obtained for the computation of
intersection numbers so far; in Section 7.4, we give a review of still open
problems plus some ideas on how to tackle them.

###  7.1 Univariate case

In this section we show how the intersection number can be evaluated exactly
in the case when integrals are defined over only one complex variable. We will
refer to the simple derivation that was presented in [76]. Let us consider a
$1$-form $\phi_{L}$ having poles at some points $z_{i}\in\mathbb{C}$. In order
to compute Eq. (246), $\iota_{\omega}(\phi_{L})$ must be constructed
explicitly. The key point lies in defining circular regions around each
$z_{i}$ point: to fix ideas, we call $V_{i}$ and $U_{i}$ two discs centered in
$z_{i}$ such that $V_{i}\subset U_{i}$; these discs are defined such that
$U_{i}\cap U_{j}=\emptyset$ for $i\neq j$. We then introduce for each $i$:

  * 1:

a holomorphic function $\psi_{i}$ such that

$\nabla_{\omega}\psi_{i}=\phi_{L}\,\,\,\text{on}\,\,\,U_{i}\setminus\left\\{z_{i}\right\\};$
(249)

  * 2:

a function $h_{i}$ such that

$h_{i}=\begin{cases}1\,\,\,\text{on}\,\,\,V_{i}\,;\\\ 0\leq h_{i}\leq
1\,\,\,\text{smooth interpolation on}\,\,\,U_{i}\setminus V_{i}\,;\\\
0\,\,\,\text{out of}\,\,\,U_{i}\,.\end{cases}$ (250)

Then $\iota_{\omega}(\phi_{L})$ can be written as

$\iota_{\omega}(\phi_{L})=\phi_{L}-\sum_{i}\nabla_{\omega}(h_{i}\psi_{i})=\phi_{L}-\sum_{i}(dh_{i}\psi_{i}+h_{i}\nabla_{\omega}\psi_{i})\,.$
(251)

Notice that in Eq. (251), as $h_{i}=0$ out of $U_{i}$, we are actually not
modifying $\phi_{L}$ in that region. Inside $V_{i}$, $h_{i}=1$, meaning that
the whole $\phi_{L}$ is subtracted and $\iota_{\omega}(\phi_{L})=0$ in the
innermost region around the singular points; on the other hand, in the outer
ring $U_{i}\setminus V_{i}$, $\phi_{L}$ is subtracted smoothly. Finally,
$-dh_{i}\psi_{i}$ is an extra term existing only in the $U_{i}\setminus V_{i}$
ring. Notice that $\phi_{L}$ and $\iota_{\omega}(\phi_{L})$ lie in the same
cohomology class, as they are identical up to a covariant derivative of a
function.
What is left now is to find the explicit form of $\psi_{i}$ obeying Eq. (249).

######  Lemma 1.

Unique existence of $\psi_{i}$.
$\exists!\,\,\psi_{i}$ such that $\psi_{i}$ is holomorphic on
$U_{i}\setminus\left\\{z_{i}\right\\}$ and $\nabla_{\omega}\psi_{i}=\phi_{L}$
on $U_{i}\setminus\left\\{z_{i}\right\\}$.

###### Proof.

Consider a 1-form $\phi_{L}$ having a pole of order $N$ at $z=z_{i}$; $\omega$
is a 1-form sharing a pole of $\phi_{L}$, $\psi_{i}$ is required to be
holomorphic: hence in terms of the local coordinate variable $z$ near $z_{i}$
(up to a change of coordinates, we can consider $z_{i}=0$ without loss of
generality) it is possible to write

$\phi_{L}=\sum_{m=-N}^{\infty}b_{m}z^{m}dz\,,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\omega=\sum_{q=-1}^{\infty}a_{q}z^{q}dz\,,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\psi_{i}=\sum_{m=-N+1}^{\infty}c_{m}z^{m}.$
(252)

Notice how in (252) $b_{-1}=\text{Res}_{z=z_{i}}(\phi_{L})$ and
$a_{-1}=\text{Res}_{z=z_{i}}(\omega)=\alpha_{i}$. Then
$\nabla_{\omega}\psi_{i}$ becomes

$\displaystyle(d+\omega\wedge)\sum_{m=-N+1}^{\infty}c_{m}z^{m}=d\left(\sum_{m=-N+1}^{\infty}c_{m}z^{m}\right)+\sum_{m=-N+1}^{\infty}\sum_{q=-1}^{\infty}a_{q}c_{m}z^{m+q}dz=$
(253)
$\displaystyle=\sum_{n=-N+1}^{\infty}\left((n+1)c_{n+1}+\sum_{q=-1}^{n}a_{q}c_{n-q}\right)z^{n}dz\,,$
(254)

in which we defined $n=m+q$: hence the sum
$\sum_{m=-N+1}^{\infty}\sum_{q=-1}^{\infty}$ must satisfy the condition
$q=n-m$, whose bigger value is obtained when $m$ is as small as possible
(which is $m=-N+1$). This implies that the sum over $q$ goes up to $n+N-1$. We
are thus lead to the identification

$(n+1)c_{n+1}+\sum_{q=-1}^{n+N-1}a_{q}c_{n-q}=b_{n}\xRightarrow{n=-1}\sum_{q=-1}^{N-2}a_{q}c_{-q-1}=b_{-1}.\qed$

The intersection number (246) can be rewritten as

$\bra{\phi_{L}}\ket{\phi_{R}}_{\omega}=\frac{1}{2\pi
i}\int_{X}\left[\phi_{L}-\sum_{i}(dh_{i})\psi_{i}-\sum_{i}h_{i}\nabla_{\omega}\psi_{i}\right]\wedge\phi_{R}=-\frac{1}{2\pi
i}\sum_{i}\int_{U_{i}\setminus V_{i}}dh_{i}\psi_{i}\wedge\phi_{R},$ (255)

where the second equality can be obtained by recalling that the first term
vanishes because $\phi_{L}\wedge\phi_{R}=0$; the second term survives where
$dh_{i}\neq 0$, i.e. in $U_{i}\setminus V_{i}$; the third term is again
proportional to $\phi_{L}\wedge\phi_{R}$ and vanishes. Notice that it holds:

$dh_{i}\psi_{i}\wedge\phi_{R}=d(h_{i}\psi_{i}\phi_{R}).$ (256)

This is because the extra terms $h_{i}d\psi_{i}\wedge\phi_{R}$ and
$h_{i}\psi_{i}d\phi_{R}$ vanish, as $d\psi_{i}$ and $\phi_{R}$ are both
holomorphic and as $\phi_{R}$ is a closed form. By the Stokes theorem, it is
possible to rewrite Eq. (255) as

$\displaystyle\bra{\phi_{L}}\ket{\phi_{R}}_{\omega}=-\frac{1}{2\pi
i}\sum_{i}\int_{U_{i}\setminus V_{i}}d(h_{i}\psi_{i}\phi_{R})=-\frac{1}{2\pi
i}\sum_{i}\int_{\partial(U_{i}\setminus
V_{i})}h_{i}\psi_{i}\phi_{R}=\frac{1}{2\pi i}\sum_{i}\int_{\partial
V_{i}}\psi_{i}\phi_{R}\,.$ (257)

Because $\partial V_{i}$ is a closed path, we can always rewrite Eq. (257) as
a sum of residues:

$\bra{\phi_{L}}\ket{\phi_{R}}_{\omega}=\sum_{i=1}^{k}\text{Res}_{z=z_{i}}(\psi_{i}\phi_{R})\,.$
(258)

###  7.2 Logarithmic $n$-forms

Evaluation of intersection numbers is particularly simple for logarithmic
forms. In the univariate case, one can check from Lemma 1 that if $\phi_{L}$
has a simple pole, than the function $\psi_{i}$ takes the form

$\psi_{i}=\frac{\text{Res}_{z=z_{i}}(\phi_{L})}{\text{Res}_{z=z_{i}}(\omega)}+\mathcal{O}(z-z_{i})=\frac{\text{Res}_{z=z_{i}}(\phi_{L})}{\alpha_{i}}+\mathcal{O}(z-z_{i})\,.$
(259)

In this case, Eq. (258) leads to the following formula for the evaluation of
the intersection number:

$\bra{\phi_{L}}\ket{\phi_{R}}_{\omega}=\sum_{i}\frac{\text{Res}_{z=z_{i}}\phi_{L}\text{Res}_{z=z_{i}}\phi_{R}}{\alpha_{i}}.$
(260)

Eq. (260) is valid for logarithmic $1$-forms, but it can be generalized to
logarithmic $n$-forms. We give only the final result for this case. Let us
consider the twisted one-form $\omega$ in the following form:

$\omega=\sum_{i=0}^{s+1}\alpha_{i}d\log f_{i}\,,$ (261)

where the $f_{i}$ are linear functions in the given variables (hence each
$f_{i}=0$ defines a certain hyperplane where the twist $\omega$ shows a
logarithmic singularity), while $\sum_{i}\alpha_{i}=0$. The cocycles
$\phi_{L,R}$ assume instead the form

$\phi_{I}=d\log\left(\frac{f_{i_{0}}}{f_{i_{1}}}\right)\wedge\cdots\wedge
d\log\left(\frac{f_{i_{n-1}}}{f_{i_{n}}}\right)$ (262)

with $I=(i_{o},\cdots,i_{n})$; the indexes are ordered such that $0\leq
i_{0}<\cdots<i_{n}\leq s+1$. In other words, $\phi_{I}$ belongs to the $n$-th
twisted cohomology group $H^{n}(X,\nabla_{\omega})$, where
$X=\mathbb{CP}^{n}\setminus\left\\{\cup_{i}{f_{i}=0}\right\\}$.
With these hypotheses, a recent work [46] has shown that the intersection
number between two logarithmic $n$-forms can also be written as

$\bra{\phi_{L}}\ket{\phi_{R}}_{\omega}=\frac{1}{(-2\pi
i)^{n}}\oint_{\wedge_{a=1}^{n}\left\\{\right|\omega_{a}|=\epsilon\\}}\frac{\phi_{L}\hat{\phi}_{R}}{\prod_{a=1}^{n}\omega_{a}},$
(263)

where $\hat{\phi}$ is the component of
$\phi=\hat{\phi}\,dz_{1}\wedge\cdots\wedge dz_{n}$ , $a$ enumerates the
dimensions involved (the number of variables) and $\omega_{a}$ represent the
components of the twist $\omega$ along each $dz_{a}$.

###  7.3 Multivariate case: recursive method

The main aim now is to be able to compute any intersection number,
independently of the order of the poles appearing and the number of dimensions
involved. A first attempt follows a recursive approach in the number of
variables (we refer to [66, 65]). In order to compute an intersection number
in $z_{1},\cdots,z_{n}$ variables, the problem can be divided into two steps:
calculating the intersection number in $z_{1},\cdots,z_{n-1}$ variables and
then a generalized intersection number depending only on $z_{n}$, which will
be introduced later. In the 1D case, $\phi_{L,R}$ and $\omega$ are 1-forms,
which we can regard as some sort of scalars. Taking into account the existence
of $\nu$ independent equivalence classes, it is possible to build vectorial
($\phi_{L,R,j})$ and tensorial ($\Omega_{ij}$) objects, with “gauge”
transformations written as

$\hat{\phi}_{L,j}^{\prime}=\hat{\phi}_{L,j}+\partial_{z}\xi_{j}+\xi_{i}\Omega_{ij}\,,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\hat{\phi}_{R,j}^{\prime}=\hat{\phi}_{R,j}+\partial_{z}\xi_{j}-\Omega_{ji}\xi_{i}\,.$
(264)

The next key idea is to study, for each $i$, the twisted cohomology group
associated to the fibration

$\nabla_{\omega}=d+\sum_{j=1}^{n}\omega_{j}dz_{j}=\sum_{j=1}^{i}dz_{j}\left(\frac{\partial}{\partial
z_{j}}+\omega_{j}\right)+\sum_{j=i+1}^{n}dz_{j}\left(\frac{\partial}{\partial
z_{j}}+\omega_{j}\right)\,,$ (265)

and define

$\omega^{(i)}:=\sum_{j=1}^{i}\omega_{j}dz_{j}\,.$ (266)

In Eq. (265) $z_{i+1},\cdots,z_{n}$ are treated as fixed parameters. Starting
from $i=n-1$, the goal is to express a cohomology class
$\bra{\phi_{L}^{(n)}}\in H_{\omega}^{n}$ using a basis of $H_{\omega}^{n-1}$,
which is supposed to be known: such basis is formed by $\nu_{n-1}$ elements
$\bra{e_{j}^{(n-1)}}$, with dual basis $\ket{d_{j}^{(n-1)}}$ such that
$\bra{e_{j}^{(n-1)}}\ket{d_{k}^{(n-1)}}=\delta_{jk}$. Inserting an identity of
the form $I=\sum_{j=1}^{\nu_{n-1}}\ket{d_{j}^{(n-1)}}\bra{e_{j}^{(n-1)}}$
leads to:

$\bra{\phi_{L}^{(n)}}=\sum_{j=1}^{\nu_{n-1}}\bra{\phi_{L,j}^{(n)}}\wedge\bra{e_{j}^{(n-1)}},\qquad\ket{\phi_{R}^{(n)}}=\sum_{j=1}^{\nu_{n-1}}\ket{d_{j}^{(n-1)}}\wedge\ket{\phi_{R,j}^{(n)}},$
(267)

where

$\bra{\phi_{L,j}^{(n)}}=\bra{\phi_{L}^{(n)}}\ket{d_{j}^{(n-1)}},\qquad\ket{\phi_{R,j}^{(n)}}=\bra{e_{j}^{(n-1)}}\ket{\phi_{R}^{(n)}}.$
(268)

Notice how the coefficients defined in Eq. (268) are not unique objects but
classes themselves. For example, the coefficient $\bra{\phi_{L,j}^{(n)}}$ is
invariant with respect to the choice within the cohomology class
$\ket{d_{j}^{(n-1)}}$ of $H_{\omega}^{*\,\,n-1}=H_{-\omega}^{n-1}$: a change
$d_{j}^{n-1}\to d_{j}^{n-1}+\nabla_{-\omega^{(n-1)}}\psi$ with $\psi$ being a
2-form would make no difference. This equivalence class is smaller than
$\bra{\phi_{L}^{(n)}}$, which is composed of all the transformations
$\phi_{L}^{(n)}\to\phi_{L}^{(n)}+\nabla_{\omega}\xi$ with $\xi$ being a
generic$(n-1)$-form: thus, the coefficients in Eq. (268) represent new
equivalence classes overall.
We introduce a new matrix $\Omega^{(n)}$ with dimension
$\nu_{n-1}\times\nu_{n-1}$

$\Omega_{ij}^{(n)}:=\bra{(\partial_{z_{n}}+\omega_{n})e_{i}^{(n-1)}}\ket{d_{j}^{(n-1)}},$
(269)

which implies

$\sum_{i}\Omega_{ij}^{(n)}\bra{e_{j}^{(n-1)}}=\bra{(\partial_{z_{n}}+\omega_{n})e_{i}^{(n-1)}}\,.$
(270)

The matrix in Eq. (269) allows to write the gauge transformations of the
coefficients in the following form (the derivation can be found in [66]):

$\hat{\phi}_{L,j}^{(n)}\to\hat{\phi}_{L,j}^{(n)}+g_{i}\left(\overleftarrow{\partial_{z_{n}}}\delta_{ij}+\Omega_{ij}^{(n)}\right)\,,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\hat{\phi}_{R,j}^{(n)}\to\hat{\phi}_{R,j}^{(n)}+\left(\delta_{ij}\overrightarrow{\partial_{z_{n}}}-\Omega_{ji}^{(n)}\right)g_{i},$
(271)

where the $g_{i}$ are arbitrary functions. With a shorter notation Eq. (271)
can be rewritten as

$\hat{\phi}_{j}\to\hat{\phi}_{j}+(\delta_{jk}\partial_{z_{n}}+\Omega_{jk})g_{k}\,,$
(272)

where the $L$ and $R$ cases are restored taking
$\Omega\equiv(\Omega^{(n)})^{\intercal}$ and $\Omega\equiv-\Omega^{(n)}$
respectively. In a number of practical cases, the matrix $\Omega^{(n)}$ can
often be chosen to have only simple poles by the application of the Moser
algorithm, whose details we do not present in this work but can be found in
[138, 139].

This property, along with Eq. (272), allows to find coefficients
$\hat{\phi}_{j}$ which are more convenient than the original ones, namely
which show only simple poles in $z_{n}$. We start looking for the
transformation which lowers the order of a pole appearing in $\hat{\phi}_{j}$.
We outline the procedure when a pole is located at a finite point. To fix
ideas, we call $q$ the irreducible polynomial appearing at the denominator of
$\hat{\phi}_{j}$, raised to a certain power $o$. The correct transformation is
given by the choice

$g_{j}(z_{n})=\frac{1}{q^{o-1}}\sum_{k=0}^{\text{deg}(q)-1}c_{jk}z_{n}^{k}\,,$
(273)

where the coefficients $c_{jk}$ have yet to be determined. With the
introduction of Eq. (273), Eq. (272) assumes the form

$\displaystyle\hat{\phi}_{j}$
$\displaystyle\to\hat{\phi}_{j}+\delta_{jk}\left[\partial_{z_{n}}\left(\frac{1}{q^{o-1}}\right)\sum_{s=0}^{\text{deg}(q)-1}c_{ks}z_{n}^{s}+\frac{1}{q^{o-1}}\sum_{s=0}^{\text{deg}(q)-1}c_{ks}\partial_{z_{n}}z_{n}^{s}\right]+{\frac{1}{q^{o-1}}\sum_{s=0}^{\text{deg}(q)-1}\Omega_{jk}c_{ks}z_{n}^{s}}$
$\displaystyle=\underbrace{\hat{\phi}_{j}-(o-1)\frac{\partial_{z_{n}}q}{q^{o}}\sum_{s=0}^{\text{deg}(q)-1}c_{js}z_{n}^{s}}_{1}+\underbrace{\frac{1}{q^{o-1}}\sum_{s=0}^{\text{deg}(q)-1}sc_{js}z_{n}^{s-1}}_{2}+\underbrace{\frac{1}{q^{o-1}}\sum_{s=0}^{\text{deg}(q)-1}\Omega_{jk}c_{ks}z_{n}^{s}}_{3}\,.$
(274)

Notice how the term labeled $1$ in Eq. (274) is the one involving the power
$q^{0}$ in the denominator: these terms (which are of the type
$z_{n}^{k}/q^{o}$ with $k\in\left\\{0,\cdots,\text{deg}(q)-1\right\\}$) can be
eliminated by solving $\text{deg}(q)$ equations. The solution of this system
determines $c_{jk}$ for a fixed $j$. Considering all the $j$ values, all the
appropriate $c_{jk}$ coefficients are determined by solving
$\nu_{n-1}\cdot\text{deg}(q)$ equations. A similar procedure exists when the
pole is not located at a finite point but at infinity.

Notice that, thanks to Eq. (267), it is possible to write the intersection
number as

$\bra{\phi_{L}}\ket{\phi_{R}}_{\omega}=\sum_{z_{0}\in
S_{n}}\sum_{j=1}^{\nu_{n-1}}\text{Res}_{z_{n}=z_{0}}\left(\hat{\psi}_{L,j}^{(n)}\bra{e_{j}^{(n-1)}}\ket{\phi_{R}}\right)\,,$
(275)

where $S_{n}$ is the set comprehensive of all the singular points of
$\Omega^{(n)}$ in $z_{n}$, while $\hat{\psi}_{L,j}^{(n)}$ is a function which
locally solves

$\partial_{z_{n}}\hat{\psi}_{L,j}^{(n)}+\hat{\psi}_{L,i}^{(n)}\Omega_{ij}^{(n)}=\hat{\phi}_{L,j}^{(n)}.$
(276)

Notice how this situation closely resembles the one outlined by Eq. (260) and
Eq. (249). Similarly to the result given by Lemma 1, the solution of Eq. (276)
is given by

$\hat{\psi}_{L,j}^{(n)}=\sum_{i}\text{Res}_{z_{n}=z_{0}}\hat{\phi}_{L,i}^{(n)}\left(\text{Res}_{z_{n}=z_{0}}\Omega^{(n)}\right)_{ij}^{-1}+\mathcal{O}(z_{n}-z_{0}).$
(277)

Notice that only the first term in Eq. (277) contributes to the residue in Eq.
(275); hence, recalling that $\hat{\phi}_{L,i}^{(n)}$ and $\Omega^{(n)}$ have
at most simple poles in $z_{n}$, it is possible to substitute

$\hat{\psi}_{L,j}^{(n)}\to\sum_{i}\hat{\phi}_{L,i}^{(n)}\left(\Omega^{(n)}\right)_{ij}^{-1}.$
(278)

Remembering that
$\text{adj}\,\,\Omega^{(n)}=\text{det}(\Omega^{(n)})\cdot(\Omega^{(n)})^{-1}=(\Omega^{(n)})^{-1}P/Q$
(we write $\det\Omega^{(n)}=P/Q$) we can finally obtain

$\displaystyle\bra{\phi_{L}}\ket{\phi_{R}}_{\omega}$
$\displaystyle=\sum_{z_{0}\in
S_{n}}\sum_{i,j=1}^{\nu_{n-1}}\text{Res}_{z_{n}=z_{0}}\left(\frac{Q}{P}\hat{\phi}_{L,i}^{(n)}\left(\text{adj}\Omega^{(n)}\right)_{ij}\hat{\phi}_{R,j}^{(n)}\right)$
$\displaystyle=\frac{1}{2\pi
i}\int_{\gamma}dz_{n}\sum_{i,j=1}^{\nu_{n-1}}\frac{Q\hat{\phi}_{L,i}^{(n)}\left(\text{adj}\Omega^{(n)}\right)_{ij}\hat{\phi}_{R,j}^{(n)}}{P}\,,$
(279)

where by $\gamma$ we mean the contour of small counterclockwise circles around
the $z_{0}\in S_{n}$ points.
Let’s take a deeper look at Eq. (279). In the univariate case, the twist is a
certain $\omega=\omega_{1}dz_{1}$ with $\omega_{1}=P/Q$ , where $P$ and $Q$
are certain polynomials in the only variable $z_{1}$. Calling
$C_{1}=\left\\{z_{1}\in\mathbb{C}:P(z_{1})=0\right\\}$ and $I_{1}=<P>$ the
ideal generated from $P$, then $C$ is the algebraic variety generated from
$I_{1}$: $C=V(I_{1})$. If $\phi_{L,R}$ have only simple poles, then we can
express the intersection number between such forms as a global residue:

$\bra{\phi_{L}}\ket{\phi_{R}}_{\omega}=-\text{Res}_{<P>}(Q\hat{\phi}_{L}\hat{\phi}_{R})\,.$
(280)

The notation in Eq. (280) is as follows [140]: given a certain form

$\tilde{\omega}=\frac{h(z)dz_{1}\wedge\cdots\wedge dz_{n}}{f_{1}(z)\cdots
f_{n}(z)},$ (281)

in which the variables are $z=(z_{1},\cdots,z_{n})$ and $f\equiv
f_{1}(z),\cdots,f_{n}(z)$ are $n$ holomorphic functions in the neighbourhood
of the closure $\overline{U}$ of the ball
$U=\left\\{||z-\xi||<\epsilon\right\\}$ which have a unique isolated zero
given by the point $\xi$ ($f^{-1}(0)=\xi)$, then the residue of
$\tilde{\omega}$ at the point $\xi$ is defined as

$\text{Res}_{\left\\{f_{1},\cdots,f_{n}\right\\},\xi}(\tilde{\omega})=\left(\frac{1}{2\pi
i}\right)^{n}\oint_{\Gamma}\frac{h(z)dz_{1}\wedge\cdots\wedge
dz_{n}}{f_{1}(z)\cdots f_{n}(z)}.$ (282)

In Eq. (282) the contour $\Gamma$ is given by
$\Gamma=\left\\{z:|f_{i}(z)=\epsilon_{i}|\right\\}$ with orientation given by
$d(\text{arg}f_{1})\wedge\cdots\wedge d(\text{arg}f_{n})\geq 0$. The subscript
$<P>$ in Eq. (280) refers to the fact that the residue in Eq. (282) does not
depend strictly on the $f$ functions in the denominator, but more in general
to the ideal they generate $<f_{1},\cdots,f_{n}>$, as we will see with more
detail later in section 7.4. Notice that Eq. (280) is equivalent to Eq. (260).
With this new formalism, Eq. (279) can be similarly rewritten as

$\bra{\phi_{L}}\ket{\phi_{R}}_{\omega}=-\text{Res}_{<P>}(Q\hat{\phi}_{L,i}(\text{adj}\,\,\Omega^{(n)})_{ij}\hat{\phi}_{R,j})\,,$
(283)

in which the contour given by the many counterclockwise circles of Eq. (279)
is now deformed into a single clockwise contour (hence the minus sign)
partially encircling a singular point, going towards infinity, coming back
from infinity, partially encircling the next singular point, etc. until a
closed path is formed. Eq. (283) generalizes Eq. (280) in many dimensions.
Notice how in Eq. (269) a generalized twist $\Omega^{(n)}$ was built, with
properties very similar to the ones of the usual twist $\omega$. Writing its
determinant $\det\Omega^{(n)}=P/Q$, one can build the set
$C_{n}=\left\\{z_{n}\in\mathbb{C}:P(z_{n})=0\right\\}$ (set of the points
where $\Omega^{(n)}$ doesn’t have full rank) and $I=<P>$, the ideal generated
by $P$: much similarly to what happened when writing the univariate Eq. (280).

In conclusion, the procedure described in this section outlines an algorithm
which, given as an input two cohomology classes $\phi_{L,R}$, computes their
intersection number recursively: starting from $i=n-1$, it expands
$\phi_{L,R}$ on the basis of $H_{\omega,-\omega}^{(n-1)}$ and computes the
matrix $\Omega^{(n)}$. By using transformations of the form (272), it
transforms the coefficient vectors $\phi_{L,R,j}$ in equivalent coefficients
but with only simple poles in $z_{n}$ and therefore can finally compute the
residue (283). The procedure is then repeated for all $i$, until one has only
an intersection number depending on the last variable.

###  7.4 Moving onwards: an open problem

Recursive approaches to the computation of intersection numbers are known, but
the time needed for the computation can be extremely long because of the many
steps composing the algorithm. A new question arises: is there any way to
compute (246) without having to introduce some type of recursivity (whether in
the number of variables, or in the reduction of the order of the poles
appearing in the twisted cocycles)? The fact that a recursive solution of the
problem exists suggests there might be some underlying property of
intersection numbers which could lead to a more fundamental method of
computation. Being able to understand this could make the computation faster,
but it would also imply getting a deeper understanding of the problem: in the
end, it would be possible to understand what Feynman integrals represent -
both in an algebraic and geometric perspective, not only by a straightforward
computation problem point of view.

As a first note, we observe that in every case computation can be carried out,
intersection numbers are always written in terms of a residue of the form Eq.
(282): we find it instructive to focus on Eq. (282) and highlight some of its
properties.

  * 1:

Non degenerate case
If $f=f_{1},\cdots,f_{n}$ is non degenerate (i.e. its Jacobian evaluated in
$0$ is $J_{f}(0)\neq 0$), then Eq. (282) can be evaluated by the introduction
of a change of variables $w:=f(z)$. Using the usual Cauchy formula leads to

$\displaystyle\left(\frac{1}{2\pi i}\right)^{n}\int_{\Gamma}\tilde{\omega}$
$\displaystyle=\left(\frac{1}{2\pi
i}\right)^{n}\int_{|w_{i}|=\epsilon_{i}}h(f^{-1}(w))\frac{dw_{1}\cdots
dw_{n}}{J_{f}(w)}\frac{1}{w_{1}}\cdots\frac{1}{w_{n}}=\frac{h(f^{-1}(0))}{J_{f}(0)}\,.$
(284)

  * 2:

$h\in\text{I}(f)$ ideal generated from the $f_{i}$
In this case the residue is 0. If, for example, $h(z)=g(z)f_{1}(z)$, then
$\tilde{\omega}$ is holomorphic in a bigger set, which we call
$U_{1}:=U\setminus(D_{2}+\cdots D_{n})$.
Then the contour $\Gamma_{1}=\left\\{z\,\,\text{such
that}\,\,|f_{1}(z_{1})|\leq\epsilon_{1},|f_{i}(z_{i})|=\epsilon_{i}\,\,\text{for}\,\,i\neq
1\right\\}$, which is an element of $U_{1}$, has a boundary
$\delta\Gamma_{1}=\pm\Gamma$: by the Stokes theorem, the residue of
$\tilde{\omega}$ along $\Gamma$ is then 0.

While the general degenerate case is hard to compute,in [141] is suggested
that if each of the $z_{i}$ appearing in $\tilde{\omega}$ depends only on a
single variable $z_{i}$, it is possible to factorize Eq. (282) as

$\text{Res}_{(0)}(\tilde{\omega})=\left(\frac{1}{2\pi
i}\right)^{n}\oint\frac{dz_{1}}{f_{1}(z_{1})}\cdots\oint\frac{dz_{n}}{f_{n}(z_{n})}h(z).$
(285)

How can this be achieved? Algebraic geometry [140] shows that actually Eq.
(282) can be reinterpreted in terms of sheaf cohomology, which allows to
obtain an important theorem:

######  Theorem 7.

Transformation Theorem
Let $f=\left\\{f_{1},\cdots,f_{n}\right\\}$ and
$g=\left\\{g_{1},\cdots,g_{n}\right\\}$ be holomorphic maps
$f,g:\bar{U}\to\mathbb{C}^{n}$ such that $f^{-1}(0)=g^{-1}(0)=0$ and
$g_{i}(z)=\sum_{j}A_{ij}(z)f_{j}(z)$, with $A_{ij}$ being a holomorphic
matrix. Then

$\textup{Res}_{(0)}\left(\frac{h(z)dz_{1}\wedge\cdots\wedge
dz_{n}}{f_{1}(z)\cdots
f_{n}(z)}\right)=\textup{Res}_{(0)}\left(\frac{h(z)dz_{1}\wedge\cdots\wedge
dz_{n}}{g_{1}(z)\cdots g_{n}(z)}\det A\right)\,.$ (286)

Given $n$ functions $g_{i}(z)=g_{i}(z_{i})$ obeying the hypotheses of Theorem
7, then the residue in Eq. (282) would take the form in Eq. (285). Calling
$R=\mathbb{K}(z_{1},\cdots,z_{n})$ the ring over a field $\mathbb{K}$ with $n$
variables $z_{1},\cdots,z_{n}$, we notice that the ideal
$I=<f_{1},\cdots,f_{n}>\subset R$ generated by the $f_{1},\cdots,f_{n}$,
defined as $I=\left\\{\sum_{i}h_{i}f_{i}\,\,|\,\,h_{i}\in R\right\\}$ implies
that the ideal $J$ generated by the $g_{i}$ polynomials is a subset of $I$:

$J=\left\\{\sum_{i}\tilde{h}_{i}g_{i}=\sum_{ij}\tilde{h}_{i}A_{ij}(z)f_{j}=\sum_{i}\left(\sum_{j}\tilde{h}_{j}A_{ji}(z)\right)f_{i}\,\,|\,\,\tilde{h}_{i}\in
R\right\\}\subset I.$ (287)

This observation implies that theorem 7 can be reinterpreted in the following
way: the residue does not depend on the specific generators of the ideal $I$:
it is possible to use the generators of an ideal which is a subset of the
original one. As suggested in [141], there is a way to generate such an ideal:
the exploitation of Gröbner bases, which can be regarded as a set of
polynomials with special properties. In Appendix B we give a brief
introduction on the topic (for more details, see for example [142]).
Going back to the problem of the computation of the residue and following the
idea from [141], we recall that the aim is to find some new
$g_{i}(z)=A_{ij}(z)f_{j}(z)$ which are more convenient than the original
$f_{i}$ polynomials. In order to find $g_{i}(z)=g_{i}(z_{i})$ for a fixed $i$,
we define a Lexicographic order in which the $i$-th variable $z_{i}$ is the
“least important variable”:

$z_{i+1}\succ z_{i+2}\succ\cdots\succ z_{n}\succ z_{1}\succ
z_{2}\succ\cdots\succ z_{i}\,.$ (288)

Starting from the original $f_{1},\cdots,f_{n}$ polynomials and constructing a
Gröbner basis while keeping this order in mind, one fact is bound to happen:
one polynomial in the basis must depend only on the $z_{i}$ variable. In fact,
the obtained Gröbner basis can be reduced with respect to itself (i.e. each
polynomial in the basis can be divided with respect to the other ones):
because of the chosen order, this process eliminates progressively the
variables $z_{1},\cdots,z_{i-1},z_{i+1},z_{n}$, leaving at least one
polynomial depending only on the last variable $z_{i}$. This polynomial is
$g_{i}(z)=g_{i}(z_{i})$. Notice that, by the Buchberger’s algorithm, the many
elements of the basis are constructed as a linear combination of the former
polynomials: all of them are of the form requested by the Transformation
Theorem hypotheses. Repeating this process for all the $n$ possible
Lexicographic orders, at the end a set
$\\{g_{1}(z_{1}),\cdots,g_{n}(z_{n})\\}$ is created. The computation of
Gröbner bases is achieved by the software Macaulay2 [143]: here we give a
brief example of how this procedure works.

######  Example 1.

As an example, we study the residue for
$\tilde{\omega}=\frac{z_{2}dz_{1}\wedge dz_{2}}{z_{1}^{2}(z_{2}-z_{1})}$.

Here we use $h(z)=z_{2}$, $f_{1}(z)=z_{1}^{2},f_{2}(z)=z_{2}-z_{1}$. The
Jacobian is degenerate in $(0,0)$, hence we compute the residue of
$\tilde{\omega}$ around $(0,0)$.
First, we look for $g_{1}(z_{1})$: in this case there is no need to start
creating a Gröbner basis associated to the ordering $z_{2}\succ z_{1}$:
$f_{1}(z)$ already depends only on $z_{1}$, hence we simply consider
$g_{1}(z_{1})=f_{1}(z)=z_{1}^{2}$.
We now switch to the problem of finding $g_{2}(z_{2})$, using the order
$z_{1}\succ z_{2}$: it is clear that it is not possible to obtain a function
depending only on $z_{2}$ dividing each $f_{i}$ by the other remaining
polynomial, so we build the Gröbner basis. Notice how this suggests that
$\left\\{f_{1},f_{2}\right\\}$ cannot form a complete Gröbner basis for the
chosen order.

$S(f_{1},f_{2})=\text{lcm}(z_{1}^{2},z_{1})\left(\frac{f_{1}}{z_{1}^{2}}+\frac{f_{2}}{z_{1}}\right)=z_{1}z_{2}.$
(289)

Dividing $S(f_{1},f_{2})$ with respect to the set
$\left\\{f_{1},f_{2}\right\\}$ with the goal to obtain a remainder which is
“smaller” in the sense of the specified order leads to

$S(f_{1},f_{2})=z_{1}z_{2}\xrightarrow{+z_{2}f_{2}}z_{2}^{2}\,,$ (290)

which is irreducible. We then add $S(f_{1},f_{2})$ (or better yet, its
remainder, which has already went through the division process) to the
original set: we obtain $\left\\{f_{1},f_{2},f_{3}\right\\}$ with
$f_{3}=z_{2}^{2}$. Without checking if this new set is a Gröbner basis or not
(which it actually is), we identify $g_{2}(z_{2})=f_{3}(z)=z_{2}^{2}$.

$\displaystyle\begin{pmatrix}g_{1}\\\
g_{2}\end{pmatrix}=\begin{pmatrix}A_{11}&A_{12}\\\
A_{21}&A_{22}\end{pmatrix}\begin{pmatrix}f_{1}\\\
f_{2}\end{pmatrix}\,\,\,\text{with}\,\,\,\begin{pmatrix}A_{11}&A_{12}\\\
A_{21}&A_{22}\end{pmatrix}=\begin{pmatrix}1&0\\\
1&z_{1}+z_{2}\end{pmatrix}\Rightarrow\det A=z_{1}+z_{2}$ (291)

By Theorem 7, the residue can be written as

$\displaystyle\text{Res}_{\left\\{f_{1},f_{2}\right\\}(0,0)}\tilde{\omega}$
$\displaystyle=\text{Res}_{\left\\{g_{1},g_{2}\right\\}(0,0)}\left(\frac{z_{2}(z_{1}+z_{2})dz_{1}\wedge
dz_{2}}{z_{1}^{2}z_{2}^{2}}\right)=\left(\frac{1}{2\pi
i}\right)^{2}\oint\frac{dz_{1}}{z_{1}^{2}}\oint
dz_{2}\frac{z_{2}(z_{2}+z_{1})}{z_{2}^{2}}=1.$ (292)

We wonder if it is always possible to write an intersection number in terms of
a residue of the kind in Eq. (282), not only when simple poles are involved.
Regarding the emerging importance of the global residue theorem in the
computation of intersection numbers, we mention a recent work [62]. It is
argued that intersection numbers between forms can be computed by an expansion
in the parameter $\gamma$ appearing in the Baikov polynomial $B^{\gamma}$,
both for big and small $\gamma$: it appears that every term in the expansion
can be written as a residue. We briefly focus on this result in order to give
an intuitive idea of the reasons behind it without entering into too much
detail.
We start by considering the manifold
$M=\mathbb{CP}^{m}\setminus\cup_{i=1^{k}}H_{i}$, where $H_{i}$ represent some
hyperplanes defined by certain equations $f_{i}=0$. With these functions it is
possible to build the holomorphic function

$W=\sum_{i=1^{k}}\alpha_{i}\log{f_{i}}\,,$ (293)

which only has logarithmic singularities along the $H_{i}$ hyperplanes. Eq.
(293) also defines a holomorphic 1-form $dW$, which in the language introduced
in Section 2 corresponds to $d\log(B)$. Similarly, it is possible to construct
the operator

$\nabla_{dW}=d+\gamma dW\wedge\,,$ (294)

which corresponds to our $d+\omega\wedge$ (remember that
$\omega=d\log(B^{\gamma})=\gamma\,d\log(B)$). We consider the space
$\Omega^{k}(M)$ of the k-forms on $M$ and construct two sequences:

$0\xrightarrow{dW\wedge}\Omega^{1}(M)\xrightarrow{dW\wedge}\cdots\xrightarrow{dW\wedge}\Omega^{m}(M)\xrightarrow{dW\wedge}0$
(295)

and

$0\xrightarrow{\nabla_{dW}}\Omega^{1}(M)\xrightarrow{\nabla_{dW}}\cdots\xrightarrow{\nabla_{dW}}\Omega^{m}(M)\xrightarrow{\nabla_{dW}}0\,.$
(296)

Notice how both sequences are well defined, as $dW\wedge dW=0$ and
$\nabla_{dW}^{2}=0$. The associated cohomology groups $H^{k}(M,dW\wedge)$ and
$H^{k}(M,\nabla_{dW})$ are respectively defined as

$H^{k}(M,dW\wedge)\coloneqq\\{\phi_{k}\in\Omega^{k}(M)|dW\wedge\phi_{k}=0\\}/\\{\phi_{k}\in\Omega^{k}(M)|\phi_{k}=dW\wedge\phi_{k-1},\text{with}\,\phi_{k-1}\in\Omega^{k-1}(M)\\}$
(297)

and

$H^{k}(M,\nabla_{dW})\coloneqq\\{\phi_{k}\in\Omega^{k}(M)|\nabla_{dW}\phi_{k}=0\\}/\\{\phi_{k}\in\Omega^{k}(M)|\phi_{k}=\nabla_{dW}\phi_{k-1},\text{with}\,\phi_{k-1}\in\Omega^{k-1}(M)\\}$
(298)

All cohomology groups are trivial for each $k\neq m$: only $H^{m}(M,dW\wedge)$
and $H^{m}(M,\nabla_{dW})$ contain useful information. We define two pairings,
the first between $H^{m}(M,dW\wedge)$ and itself and the second between
$H^{m}(M,\nabla_{dW})$ and $H^{m}(M,\nabla_{-dW})$. In the first case, taking
$\phi_{\pm}\in H^{m}(M,\nabla_{dW})$ it is possible to define [46]

$(\phi_{-}|\phi_{+})_{dW,0}=\text{Res}_{dW=0}\left(\frac{\hat{\phi}_{-}\hat{\phi}_{+}d^{m}z}{\partial_{1}W\cdots\partial_{m}W}\right)\,,$
(299)

where the $\,\hat{}\,$ notation strips the forms from their $d^{m}z$ part and
where $\text{Res}_{dW=0}$ represents a sum of residues around each critical
point given by $\partial_{i}W=0$, for $i=1,\cdots,m$.
In the second case, taking $\phi_{\pm}\in H^{m}(M,\nabla_{\pm dW})$ allows to
define

$\bra{\phi_{-}}\ket{\phi_{+}}_{dW}=\left(\frac{\gamma}{2\pi
i}\right)^{m}\int_{M}\phi_{-}\wedge\phi_{+}^{c}\,,$ (300)

where $\phi_{+}^{c}$ represents the compact version of $\phi_{+}$ belonging to
the same equivalence class as $\phi_{+}$, similarly to the usual definitions
from Cho and Matsumoto [74] (in Eq. (246) we take the compact version of
$\phi_{-}$, but it is just a matter of convention): Eq. (300) represents an
intersection number. It is argued that, like (297) looks like a limit for big
$\gamma$ of (298), then the pairings in Eq. (299) and Eq. (300) must be
linked. It is found that Eq. (300) can be expanded as follows:

$\bra{\phi_{-}}\ket{\phi_{+}}_{dW}=\sum_{k=0}^{\infty}\gamma^{-k}(\phi_{-}|\phi_{+})_{dW,k}\,$
(301)

where each $(\phi_{-}|\phi_{+})_{dW,k}$ is a higher residue pairing [144] and
has the form of a $\text{Res}_{dW=0}$ of certain functions which only contain
$\hat{\phi}_{-}$, $\hat{\phi}_{+}$ in the numerator and the derivatives
$\partial_{i}W$ both in numerator and denominator.

Notice that, as it can be seen in Example 1, any residue of the kind we met
can be calculated independently of the order of the poles appearing. If - as
suggested by [62] \- intersection numbers were always found to be expressed
only in terms of residues, then a procedure involving Gröbner bases could be
exploited. The biggest difference between the residue in Eq. (282) and the
integral in Eq. (3) is the type of cohomology involved. Consider a
$(n-1)$-form $\tilde{\xi}$ of the same type as in Eq. (281):

$\tilde{\xi}=\frac{\tilde{h}(z)dz_{1}\wedge\cdots\wedge
dz_{n-1}}{\tilde{f}_{1}(z)\cdots\tilde{f}_{n-1}(z)}.$ (302)

The form $\tilde{\xi}$ is holomorphic everywhere in
$U=\left\\{z\in\mathbb{C}^{n-1}\,\,\text{such
that}\,\,||z||<\epsilon\right\\}$ except from the set $\tilde{D}$, defined as
$\tilde{D}=\tilde{D}_{1}+\cdots+\tilde{D}_{n-1}$, where each
$\tilde{D}_{i}=(\tilde{f}_{i})$ is the divisor of $\tilde{f}_{i}$ (namely, it
is the formal sum of the points sent to 0 by applying $\tilde{f}_{i}$). This
implies that its external derivative is null: $d\tilde{\xi}=0$, so that
$\int_{\Gamma}d\tilde{\xi}=0$, where $\Gamma$ is the contour appearing in Eq.
(282). As the addition of a term of the type $d\tilde{\xi}$ gives no
contribution to Eq. (282), the residue of $\tilde{\omega}$ depends on the de
Rham cohomology class $[\tilde{\omega}]\in\text{H}^{n}_{\text{dR}}(U\setminus
D)$, where $D=D_{1}+\cdots+D_{n}$, with $D_{i}=(f_{i})$. On the other side,
the integral (3) similarly depends on the twisted cohomology class of $\phi$.
This difference is due to the presence of $u=B^{\gamma}$ inside the integral:
as we have seen earlier, the effect of $u$ is the introduction of some cuts in
the complex space: we wonder if, in a sense, $u$ can be considered as part of
the geometry and can be “transferred inside the sign of integral”. The
integral $I$ on the cut complex space can also be thought as an integral on a
certain Riemann surface where the cuts have been glued together. We wonder if,
in this sense, it is possible to rewrite the integral (3) as multi-variable
residue with a form similar to Eq. (282), or at least as a sum of terms of
this kind as suggested in [62]. In this case, a similar approach could be
followed: while Theorem 7 emerges from the interpretation of the residue by
means of sheaf cohomology - in which the cohomology of interest is the de Rham
cohomology - another similar theorem for the twisted cohomology group should
exist, telling how one can adapt the function inside the integral while
obtaining the same residue. As a first guess, it shouldn’t differ too much
from the original theorem. Once having derived how the integrand can be
modified to obtain a more pleasing integral, the focus could remain the same:
having functions in the denominator which depend only from a single variable,
so that the computation of the integral can finally be performed.

Of course, while we stress that these observations only outline a possible way
to tackle the problem of computing an intersection number as a whole, we
highlight how certainly the path has to lie in looking at Eq. (246) not only
with straightforward computation in mind, but with the objective to understand
a deeper structure: the path certainly lies in the abstract procedure outlined
in Section 6. In this section, we introduced many of the known practical
tools: some have already been exploited for the actual evaluation of
intersection numbers (Sections 7.1-7.3), while in Section 7.4 we introduced
some concepts and definitions which may become part of the practical
instruments needed to pursue, in the end, the computation of the Feynman
integral.

##  8 An explicit example of Feynman integral

We want to conclude this review by showing how to explicitly apply
intersection theory to the computation of Feynman integrals. The present
example is due to Manoj Mandal and Federico Gasparotto. Let us consider the
double box diagram for massless particles:

$\boldsymbol{p_{1}}$$\boldsymbol{p_{2}}$$\boldsymbol{p_{3}}$$\boldsymbol{p_{4}}$$\boldsymbol{k_{1}}$$\boldsymbol{k_{2}}$$\boldsymbol{k_{1}-p_{1}-p_{2}}$$\boldsymbol{k_{2}-p_{1}-p_{2}}$$\boldsymbol{k_{1}-p_{1}}$$\boldsymbol{k_{1}-k_{2}}$$\boldsymbol{k_{2}-p_{1}-p_{2}-p_{3}}$
Figure 23: Double box diagram.

After reducing to the maximal cut and choosing a basis of Master Integrals, we
will show how to use intersection theory for determining the differential
equations for the MIs as functions of a Mandelstam’s variable. Of course,
momentum conservation requires $p_{4}=-p_{1}-p_{2}-p_{3}$, so we have $E=3$
independent external momenta, while $L=2$, so we need

$\displaystyle M=EL+\frac{L(L+1)}{2}=9$ (303)

Baikov’s variables. Seven are provided by the true denominators

$\displaystyle D_{1}$ $\displaystyle=k_{1}^{2},\qquad\
D_{2}=(k_{1}-p_{1})^{2},\qquad\ D_{3}=(k_{1}-p_{1}-p_{2})^{2},\qquad
D_{4}=(k_{1}-k_{2})^{2},$ (304) $\displaystyle D_{5}$
$\displaystyle=(k_{2}-p_{1}-p_{2})^{2},\qquad\
D_{6}=(k_{2}-p_{1}-p_{2}-p_{3})^{2},\qquad\ D_{7}=k_{2}^{2},$ (305)

while the remaining two are given by the fake denominators

$\displaystyle D_{8}$ $\displaystyle=(k_{2}-p_{1})^{2},\qquad\
D_{9}=(k_{1}-p_{1}-p_{2}-p_{3})^{2}.$ (306)

Using the Mandelstam’s variables111111Notice also that the massless condition
and momentum conservation imply $s+t=-2p_{1}\cdot p_{3}$

$\displaystyle s=2p_{1}\cdot p_{2},\qquad t=2p_{2}\cdot p_{3},$ (307)

the Baikov polynomials $B$ in the variables $z_{j}=D_{j}$, $j=1,\ldots,9$, is
given by

$\displaystyle\frac{16B}{(-st(s+t))^{\frac{4-d}{2}}}=$ $\displaystyle
st^{2}z_{1}z_{3}+s^{2}t^{2}z_{4}-st^{2}z_{1}z_{4}-st^{2}z_{2}z_{4}-st^{2}z_{3}z_{4}+(s^{2}t+st^{2})z_{4}^{2}-st^{2}z_{1}z_{5}+t^{2}z_{1}^{2}z_{5}+stz_{1}z_{2}z_{5}$
(308)
$\displaystyle-t^{2}z_{1}z_{3}z_{-}st^{2}z_{4}z_{5}-2stz_{1}z_{4}z_{5}-t^{2}z_{1}z_{4}z_{5}+stz_{2}z_{4}z_{5}+t^{2}z_{3}z_{4}z_{5}+t^{2}z_{1}z_{5}^{2}+tz_{1}^{2}z_{5}^{2}$
(309) $\displaystyle-
tz_{1}z_{2}z_{5}^{2}-s^{2}tz_{2}z_{6}+stz_{1}z_{2}z_{6}+s^{2}z_{2}^{2}z_{6}-2stz_{1}z_{3}z_{6}+stz_{2}z_{3}z_{6}-s^{2}tz_{4}z_{6}+stz_{1}z_{4}z_{6}$
(310)
$\displaystyle-s^{2}z_{2}z_{4}z_{6}-2stz_{2}z_{4}z_{6}+stz_{3}z_{4}z_{6}+stz_{1}z_{5}z_{6}-tz_{1}^{2}z_{5}z_{6}+stz_{2}z_{5}z_{6}+(s+t)z_{1}z_{2}z_{5}z_{6}$
(311) $\displaystyle-
sz_{2}^{2}z_{5}z_{6}+tz_{1}z_{3}z_{5}z_{6}-tz_{2}z_{3}z_{5}z_{6}+s^{2}z_{2}z_{6}^{2}-sz_{1}z_{2}z_{6}^{2}+sz_{2}^{2}z_{6}^{2}+sz_{1}z_{3}z_{6}^{2}$
(312) $\displaystyle-
sz_{2}z_{3}z_{6}^{2}-st^{2}z_{3}z_{7}-t^{2}z_{1}z_{3}z_{7}+stz_{2}z_{3}z_{7}+t^{2}z_{3}^{2}z_{7}-st^{2}z_{4}z_{7}+t^{2}z_{1}z_{4}z_{7}+stz_{2}z_{4}z_{7}$
(313)
$\displaystyle-2stz_{3}z_{4}z_{7}-t^{2}z_{3}z_{4}z_{7}+st^{2}z_{5}z_{7}-t^{2}z_{1}z_{5}z_{7}-2stz_{2}z_{5}z_{7}+tz_{1}z_{2}z_{5}z_{7}+sz_{2}^{2}z_{5}z_{7}$
(314)
$\displaystyle-t^{2}z_{3}z_{5}z_{7}-2tz_{1}z_{3}z_{5}z_{7}+tz_{2}z_{3}z_{5}z_{7}+stz_{2}z_{6}z_{7}-tz_{1}z_{2}z_{6}z_{7}-sz_{2}^{2}z_{6}z_{7}+stz_{3}z_{6}z_{7}$
(315)
$\displaystyle+tz_{1}z_{3}z_{6}z_{7}+(s+t)z_{2}z_{3}z_{6}z_{7}-tz_{3}^{2}z_{6}z_{7}+t^{2}z_{3}z_{7}^{2}-tz_{2}z_{3}z_{7}^{2}+tz_{3}^{2}z_{7}^{2}-2stz_{1}z_{3}z_{8}$
(316)
$\displaystyle-s^{2}tz_{4}z_{8}+stz_{1}z_{4}z_{8}+s^{2}z_{2}z_{4}z_{8}+stz_{3}z_{4}z_{8}+stz_{1}z_{5}z_{8}-tz_{1}^{2}z_{5}z_{8}-sz_{1}z_{2}z_{5}z_{8}$
(317)
$\displaystyle+tz_{1}z_{3}z_{5}z_{8}+s^{2}tz_{6}z_{8}-2stz_{1}z_{6}z_{8}+tz_{1}^{2}z_{6}z_{8}-s^{2}z_{2}z_{6}z_{8}+sz_{1}z_{2}z_{6}z_{8}-2stz_{3}z_{6}z_{8}$
(318)
$\displaystyle-2(s+t)z_{1}z_{3}z_{6}z_{8}-sz_{2}z_{3}z_{6}z_{8}+tz_{3}^{2}z_{6}z_{8}+stz_{3}z_{7}z_{8}+tz_{1}z_{3}z_{7}z_{8}-sz_{2}z_{3}z_{7}z_{8}$
(319) $\displaystyle-
tz_{3}^{2}z_{7}z_{8}+sz_{1}z_{3}z_{8}^{2}+s^{2}tz_{2}z_{9}-s^{2}tz_{4}z_{9}+stz_{1}z_{5}z_{9}-2stz_{2}z_{5}z_{9}+stz_{4}z_{5}z_{9}$
(320) $\displaystyle-
tz_{1}z_{5}^{2}z_{9}+tz_{2}z_{5}^{2}z_{9}-s^{2}z_{2}z_{6}z_{9}+s^{2}z_{4}z_{6}z_{9}-sz_{1}z_{5}z_{6}z_{9}+sz_{2}z_{5}z_{6}z_{9}-2stz_{2}z_{7}z_{9}$
(321)
$\displaystyle+stz_{3}z_{7}z_{9}+stz_{4}z_{7}z_{9}-2stz_{5}z_{7}z_{9}+tz_{1}z_{5}z_{7}z_{9}-2(s+t)z_{2}z_{5}z_{7}z_{9}+tz_{3}z_{5}z_{7}z_{9}+sz_{2}z_{6}z_{7}z_{9}$
(322) $\displaystyle-
sz_{3}z_{6}z_{7}z_{9}+tz_{2}z_{7}2z_{9}-tz_{3}z_{7}^{2}z_{9}-s^{2}tz_{8}z_{9}+stz_{1}z_{8}z_{9}-s^{2}z_{2}z_{8}z_{9}+stz_{3}z_{8}z_{9}$
(323)
$\displaystyle-s^{2}z_{4}z_{8}z_{9}-2stz_{4}z_{8}z_{9}+stz_{5}z_{8}z_{9}+sz_{1}z_{5}z_{8}z_{9}+tz_{1}z_{5}z_{8}z_{9}+sz_{2}z_{5}z_{8}z_{9}-tz_{3}z_{5}z_{8}z_{9}$
(324)
$\displaystyle-s^{2}z_{6}z_{8}z_{9}+sz_{1}z_{6}z_{8}z_{9}-2sz_{2}z_{6}z_{8}z_{9}+sz_{3}z_{6}z_{8}z_{9}+stz_{7}z_{8}z_{9}-tz_{1}z_{7}z_{8}z_{9}+sz_{2}z_{7}z_{8}z_{9}$
(325)
$\displaystyle+(s+t)z_{3}z_{7}z_{8}z_{9}+s^{2}z_{8}^{2}z_{9}-sz_{1}z_{8}^{2}z_{9}-sz_{3}z_{8}^{2}z_{9}+sz_{5}z_{7}z_{9}^{2}+s^{2}z_{8}z_{9}^{2}-sz_{5}z_{8}z_{9}^{2}$
(326) $\displaystyle-sz_{7}z_{8}z_{9}^{2}+sz_{8}^{2}z_{9}^{2}.$ (327)

However, we will not work with the complete diagram but on the maximal cut,
which correspond to set $z_{j}=0$, $j=1,\ldots,7$ so that only the
denominators $D_{8}$ and $D_{9}$ survive. Using $w_{1}=z_{8}$ and
$w_{2}=z_{9}$ we then get for the maximal cut

$\displaystyle
B(w_{1},w_{2})=\frac{(-st(s+t))^{\frac{4-d}{2}}}{16}\left[-s^{2}tw_{1}w_{2}+s^{2}w_{1}^{2}w_{2}+s^{2}w_{1}w_{2}^{2}+sw_{1}^{2}w_{2}^{2}\right].$
(328)

The twisting section is thus

$\displaystyle u=B^{\gamma}$ (329)

with

$\displaystyle\gamma=\frac{d-E-L-1}{2}=\frac{d-6}{2}.$ (330)

The associated connection 1-form is

$\displaystyle\omega=\gamma\frac{-st+2sw_{1}+sw_{2}+2w_{1}w_{2}}{-stw_{1}+sw_{1}^{2}+sw_{1}w_{2}+w_{1}^{2}w_{2}}dw_{1}+\gamma\frac{-st+sw_{1}+2sw_{2}+2w_{1}w_{2}}{-stw_{2}+sw_{1}w_{2}+sw_{2}^{2}+w_{1}w_{2}^{2}}dw_{2}.$
(331)

The equation $\omega=0$ has two solutions

$\displaystyle
w_{1}=w_{2}=w_{\pm}\equiv-\frac{3}{4}s\pm\sqrt{\frac{9}{16}s^{2}+\frac{1}{2}st},$
(332)

so there are two MIs. Since we have to compute bivariate intersection numbers,
we consider $w_{1}$ as internal integration variable. The number of
corresponding MIs is given by the solutions of $\omega_{1}=0$ for $w_{1}$,
where $\omega=\omega_{1}dw_{1}+\omega_{2}dw_{2}$. Therefore, we need just one
MI for the internal integration.
As MIs we choose $I_{[1,1,1,1,1,1,1,0,0]}$ and $I_{[1,1,1,1,1,1,1,-1,0]}$,
which correspond to the forms

$\displaystyle\bra{e_{1}}\equiv\bra{1},\qquad\bra{e_{2}}\equiv\bra{w_{1}},$
(333)

respectively. Considering them as functions of the external parameter $s$,
they have to satisfy the differential equation system

$\displaystyle\bra{\partial_{s}e_{j}+(\partial_{s}\log
u)e_{j}}=\sum_{k=1}^{2}O_{jk}\bra{e_{k}}$ (334)

where the matrix $\boldsymbol{O}$ is obtained as follows (see [61], Section
3.2). Let $\phi_{sj}:=\partial_{s}e_{j}+(\partial_{s}\log u)e_{j}$,
$F_{jk}=\bra{\phi_{sj}}d_{k}\rangle$ and $C_{jk}=\bra{e_{j}}d_{k}\rangle$
define the matrices $\boldsymbol{F}$ and $\boldsymbol{C}$. Then,

$\displaystyle\boldsymbol{O}=\boldsymbol{F}\boldsymbol{C}^{-1}.$ (335)

We will then show how to compute $\boldsymbol{F}$ and $\boldsymbol{C}$, being
very explicit for $C_{11}$ and quoting the results for all the other
components. Notice that in our specific case $\partial_{s}e_{j}=0$ and

$\displaystyle\boldsymbol{\phi}_{s}=\frac{\gamma}{s}\begin{pmatrix}\frac{2s(t-w_{1}-w_{2})-w_{1}w_{2}}{s(t-w_{1}-w_{2})-w_{1}w_{2}}\\\
\frac{w_{1}(2s(t-w_{1}-w_{2})-w_{1}w_{2})}{s(t-w_{1}-w_{2})-w_{1}w_{2})}\end{pmatrix}.$
(336)

We choose $d_{j}=e_{j}$.

###  8.1 Computation of $\boldsymbol{C}$

Let us start computing $\bra{1}1\rangle$ very explicitly. According to the
general construction, we need to start with the internal integration,
therefore we choose an internal basis (for $w_{1}$ only). Recalling that the
dimension is 1, we choose

$\displaystyle\bra{\varepsilon}=\bra{1/w_{1}}$ (337)

and $\delta=\varepsilon$ for the dual. The critical point of $d_{w_{1}}\log u$
is

$\displaystyle w_{1}=\bar{w}\equiv\frac{s}{2}\frac{t-w_{2}}{s+w_{2}}.$ (338)

Since $\varepsilon$ is a $d\log$-form, the reduced intersection matrix
$\boldsymbol{C}_{red}\equiv\bra{\varepsilon}\delta\rangle$ can be computed
with the simple method of univariate intersection numbers for $d\log$-forms.
From (263) we get

$\displaystyle\boldsymbol{C}_{red}=-{\rm
Res}_{w_{1}=\bar{w}}\left(\frac{1}{w_{1}^{2}}\frac{1}{\omega_{1}}\right)=-\frac{1}{\bar{w}^{2}}\frac{1}{\partial_{w_{1}}\omega_{1}|_{w_{1}=\bar{w}}}=\frac{1}{2\gamma}.$
(339)

To proceed iteratively, as we have seen for the recursive calculation of the
multivariate intersection numbers, we have now to determine the new connection
(269), which is now121212Notice that we are not using an orthonormal frame, so
the inverse matrix $\boldsymbol{C}_{red}^{-1}$ appears.

$\displaystyle\Omega^{(2)}=\bra{\partial_{w_{2}}\varepsilon+\omega_{2}\varepsilon}\delta\rangle\boldsymbol{C}_{red}^{-1}.$
(340)

Using again (263), we get

$\displaystyle\Omega^{(2)}=-2\gamma{\rm
Res}_{w_{1}=\bar{w}}\left(\frac{\omega_{2}}{w_{1}^{2}}\frac{1}{\omega_{1}}\right)=\frac{\gamma}{w_{2}}-\frac{\gamma}{w_{2}+s}+\frac{2\gamma}{w_{2}-t}.$
(341)

Notice that $\Omega^{(2)}$ has only simple poles and its zeros are given by
(332).
Next, we need to project $\bra{\phi_{L}}\equiv\bra{e_{1}}=\bra{1}$ to
$\bra{\varepsilon}$:

$\displaystyle\bra{\phi_{L}^{(2)}}:=\bra{\phi_{L}}\delta\rangle\boldsymbol{C}_{red}^{-1}.$
(342)

This computation is not so direct as the previous ones, since $\phi_{L}$ is
not of $d\log$-type, it has a double pole at infinity: for $w_{1}\to 1/w_{1}$
we have $\bra{1}\to\bra{-1/w_{1}^{2}}$. However, we can use the cohomological
property of the intersection numbers and shift $\phi_{L}$ by a
$\nabla_{\omega_{1}}$-exact form to get a representative in the same class
with only simple poles. For example

$\displaystyle\tilde{\phi}_{L}=\phi_{L}-\nabla_{\omega_{1}}\frac{w_{1}}{1+2\gamma}=\frac{s\gamma}{1+2\gamma}\frac{w_{2}-t}{-st+sw_{1}+sw_{2}+w_{1}w_{2}}$
(343)

has two simple poles, one at $w_{1}=\infty$ and the other one at

$\displaystyle w_{1}=s\frac{t-w_{2}}{s+w_{1}}.$

Since cohomologous formes give equal intersection numbers, we can write

$\displaystyle\bra{\phi_{L}^{(2)}}=\bra{\tilde{\phi}_{L}}\delta\rangle\boldsymbol{C}_{red}^{-1}$
(344)

and use once again (263) to get

$\displaystyle\bra{\phi_{L}^{(2)}}=-2\gamma{\rm
Res}_{w_{1}=\bar{w}}\left(\frac{\tilde{\phi}_{L}}{w_{1}}\frac{1}{\omega_{1}}\right)=\frac{s\gamma}{1+2\gamma}\frac{t-w_{2}}{s+w_{2}}.$
(345)

In a similar way, we have to project $\ket{\phi_{R}}\equiv\ket{e_{1}}=\ket{1}$
on $\ket{\delta}$:

$\displaystyle\ket{\phi_{R}^{(2)}}:=\boldsymbol{C}_{red}^{-1}\langle\varepsilon\ket{\phi_{R}}.$
(346)

As before, $\phi_{R}$ has a double pole at infinity so we replace it with the
representative131313notice that we have changed the connection to
$-\omega_{1}$

$\displaystyle\tilde{\phi}_{R}=\phi_{L}-\nabla_{-\omega_{1}}\frac{w_{1}}{1-2\gamma}=\frac{s\gamma}{(2\gamma-1)}\frac{w_{2}-t}{-st+sw_{1}+sw_{2}+w_{1}w_{2}},$
(347)

to get

$\displaystyle\ket{\phi_{R}^{(2)}}=-2\gamma{\rm
Res}_{w_{1}=\bar{w}}\left(\frac{\varepsilon\tilde{\phi}_{R}}{\omega_{1}}\right)=-\frac{s\gamma}{1-2\gamma}\frac{t-w_{2}}{s+w_{2}}.$
(348)

Before considering the final step of the computation of $C_{11}$, which
involves the intersection between $\phi_{L}^{(2)}$ and $\phi_{R}^{(2)}$
twisted with $\Omega^{(2)}$, we notice that the lasts two have again double
pole at infinity. Once again, we change the cohomology representatives in
order to work with forms having only simple poles. Since

$\displaystyle\phi_{L}^{(2)}=-\frac{s\gamma}{1+2\gamma}+\frac{\gamma
s(s+t)}{(1+2\gamma)(s+2w_{2})},$ (349)

we conveniently define

$\displaystyle\tilde{\phi}_{L}^{(2)}=\phi_{L}^{(2)}-\nabla_{\Omega^{(2)}}\left(-\frac{s\gamma}{1+2\gamma}\frac{w_{2}}{1+2\gamma}\right)=\frac{2st\gamma^{2}}{(1+2\gamma)^{2}(w_{2}-t)}+\frac{\gamma
s(s+t)+3s^{2}\gamma^{2}+2\gamma^{2}st}{(1+2\gamma)^{2}(s+w_{2})}.$ (350)

Similarly we set

$\displaystyle\tilde{\phi}_{R}^{(2)}=\phi_{R}^{(2)}-\nabla_{-\Omega^{(2)}}\left(\frac{s\gamma}{1-2\gamma}\frac{w_{2}}{1-2\gamma}\right)=\frac{2st\gamma^{2}}{(1-2\gamma)^{2}(w_{2}-t)}+\frac{-\gamma
s(s+t)+3s^{2}\gamma^{2}+2\gamma^{2}st}{(1-2\gamma)^{2}(s+w_{2})}.$ (351)

We can finally use (279) to compute the final step141414once again taking into
account of the non othonormality through $\boldsymbol{C}_{red}$

$\displaystyle\boldsymbol{C}_{11}=$ $\displaystyle\bra{1}1\rangle=-{\rm
Res}_{w_{2}=w_{+}}\left(\frac{\tilde{\phi}_{L}^{(2)}\tilde{\phi}_{R}^{(2)}}{\Omega^{(2)}}\boldsymbol{C}_{red}\right)-{\rm
Res}_{w_{2}=w_{-}}\left(\frac{\tilde{\phi}_{L}^{(2)}\tilde{\phi}_{R}^{(2)}}{\Omega^{(2)}}\boldsymbol{C}_{red}\right)$
(352) $\displaystyle=$ $\displaystyle
s^{2}\frac{3(1-8\gamma^{2})(s+t)^{2}-3\gamma^{2}s^{2}+4\gamma^{2}t^{2}}{4(1-4\gamma^{2})^{2}}.$
(353)

In the same way we can compute the remaining matrix elements of which we just
quote the final results

$\displaystyle\boldsymbol{C}_{12}=$
$\displaystyle\bra{1}w_{1}\rangle=\frac{s^{2}}{8(1-\gamma)(1-4\gamma^{2})^{2}}\left[2t^{3}\gamma(1-4\gamma^{2})-2st^{2}(3-7\gamma-22\gamma^{2}+44\gamma^{3})\right.$
(354)
$\displaystyle\left.-3s^{2}t(4-7\gamma-33\gamma^{2}+54\gamma^{3})-s^{3}(6-9\gamma-54\gamma^{2}+81\gamma^{3})\right],$
(355) $\displaystyle\boldsymbol{C}_{21}=$
$\displaystyle\bra{w_{1}}1\rangle=-\frac{s^{2}}{8(1+\gamma)(1-4\gamma^{2})^{2}}\left[2t^{3}\gamma(1-4\gamma^{2})+2st^{2}(3+7\gamma-22\gamma^{2}-44\gamma^{3})\right.$
(356)
$\displaystyle\left.+3s^{2}t(4+7\gamma-33\gamma^{2}-54\gamma^{3})+s^{3}(6+9\gamma-54\gamma^{2}-81\gamma^{3})\right]=\boldsymbol{C}_{12}(\gamma\to-\gamma),$
(357) $\displaystyle\boldsymbol{C}_{22}=$
$\displaystyle\bra{w_{1}}w_{1}\rangle=\frac{s^{3}}{16(1-4\gamma^{2})^{2}(1-\gamma^{2})}\left[-16t^{3}\gamma^{2}(1-4\gamma^{2})+12st^{2}(1-13\gamma^{2}+30\gamma^{4})\right.$
(358)
$\displaystyle\left.+12s^{2}t(2-23\gamma^{2}+45\gamma^{4})+3s^{3}(4-45\gamma^{2}+81\gamma^{4})\right].$
(359)

###  8.2 Computation of $\boldsymbol{F}$

This is computed exactly in the same way, now choosing $\phi_{L}=\phi_{sj}$
and $\phi_{R}=d_{k}$. Again, we will not repeat all passages here, but limit
ourselves to quote the final results. Of course, the interested reader is
invited to reproduce all the details carefully. We get

$\displaystyle\boldsymbol{F}_{11}=$
$\displaystyle\bra{\phi_{s1}}1\rangle=\frac{s}{4(1-4\gamma^{2})^{2}}\left[3s^{2}(2+5\gamma-18\gamma^{2}-45\gamma^{3})+t^{2}(3+11\gamma-20\gamma^{2}-68\gamma^{3})\right.$
(360) $\displaystyle\left.+3st(3+9\gamma-24\gamma^{2}-68\gamma^{3})\right],$
(361) $\displaystyle\boldsymbol{F}_{12}=$
$\displaystyle\bra{\phi_{s1}}w_{1}\rangle=\frac{s}{8(1-\gamma)(1-4\gamma^{2})^{2}}\left[2t^{3}\gamma(1+3\gamma-4\gamma^{2}-12\gamma^{3})-3s^{3}(4+4\gamma-51\gamma^{2}-36\gamma^{3}+135\gamma^{4})\right.$
(362)
$\displaystyle\left.-2st^{2}(3+3\gamma-50\gamma^{2}-26\gamma^{3}+160\gamma^{4})-3s^{2}t(6+7\gamma-82\gamma^{2}-57\gamma^{3}+234\gamma^{4})\right],$
(363) $\displaystyle\boldsymbol{F}_{21}=$
$\displaystyle\bra{\phi_{s2}}1\rangle=\frac{s}{8(1+\gamma)(1-4\gamma^{2})^{2}}\left[2t^{3}\gamma(-1-3\gamma+4\gamma^{2}+12\gamma^{3})+4st^{2}(-3-12\gamma+8\gamma^{2}+79\gamma^{3}+80\gamma^{4})\right.$
(364)
$\displaystyle\left.+3s^{3}(-6-19\gamma+39\gamma^{2}+171\gamma^{3}+135\gamma^{4})+3s^{2}t(-10-35\gamma+50\gamma^{2}+273\gamma^{3}+234\gamma^{4})\right],$
(365) $\displaystyle\boldsymbol{F}_{22}=$
$\displaystyle\bra{\phi_{s2}}w-1\rangle=\frac{s^{2}}{16(1-\gamma)(1-4\gamma^{2})^{2}}\left[4t^{3}\gamma(-1-6\gamma-9\gamma^{2}+24\gamma^{3}+52\gamma^{4})\right.$
(366)
$\displaystyle+12st^{2}(2+3\gamma-26\gamma^{2}-45\gamma^{3}+60\gamma^{4}+114\gamma^{5})+6s^{2}t(10+17\gamma-115\gamma^{2}-197\gamma^{3}+225\gamma^{4}+396\gamma^{5})$
(367)
$\displaystyle\left.+3s^{3}(12+20\gamma-135\gamma^{2}-225\gamma^{3}+243\gamma^{4}+405\gamma^{5})\right].$
(368)

###  8.3 Computation of $\boldsymbol{O}$

With these elements we can finally compute the matrix
$\boldsymbol{O}=\boldsymbol{F}\boldsymbol{C}^{-1}$ defining the differential
equation for the MIs:

$\displaystyle\boldsymbol{O}=\frac{1}{s(s+t)}\begin{pmatrix}-(2s+(d-6)t)&4-d\\\
(4-d)\frac{t}{2}&(-16+3d)\frac{s}{2}+(-6+d)t\end{pmatrix}.$ (369)

Of course we could proceed working out the present example, but our scope,
illustrating how to apply intersection theory to Feynman calculations, is
already reached, so we leave further computation to the interested readers. Of
course, exploiting all these calculations by hand is lengthy and requires a
lot of time, but the huge advantage of these methods is that they can be
implemented on a computer making it very fast.

##  9 Final comments

The strategy we followed here is rising growing interest at the time we are
writing the present review. The ability of recognizing the correct cohomology
underlying the Feynman Integrals and the corresponding intersection product
would allow both to gain deeper understanding of such integrals and to
systematize their computation. In particular, after determining the cohomology
space, one is left with the choice of a convenient basis, i.e. the Master
Integrals, with respect to which one can project the given amplitude, through
the intersection product.

In Section 6.3 we argued the necessity of introducing the vector space
$IH_{M}^{(p)}$ of twisted cohomology with perversity, in order to correctly
deal with singular forms; in Section 3 we showed different ways in which the
dimension $\nu$ of such space can be computed and interpreted. Such
interpretations mostly rely on the Poincaré duality between cycles and
cocycles, which in the case of singular varieties is restored by the
introduction of perverse sheaves. While in principle any set of $\nu$
independent forms in $IH_{M}^{(p)}$ could serve as a basis, the determination
of a preferred one suitable for practical computation is still an open
problem. We remark that a natural basis of cycles is represented by the
Lefschetz thimbles, as we described at the end of Section 6.4. This allows in
principle to define a basis also for the cohomology, thanks to Poincaré
duality. Here our excursus ends: the practical realization of such duality has
yet to be performed in a form apt to get a systematic approach in the problem
of the calculation of Feynman Integrals.

## Acknowledgements

First of all, we are indebted with professor Pierpaolo Mastrolia and the group
of Padua (Manoj Mandal, Federico Gasparotto, Luca Mattiazzi, Vsevolod
Chestnov, Hjalte Frellesvig & Co.) for introducing us to the topic of
Intersection theory methods in the Feynman integrals landscape. We also thank
Pierpaolo for helping us in improving considerably our manuscript and adding
content to it. We also thank Maxim Kontsevich for his lesson related to
Section 6.4, and Thibault Damour for helpful discussions. In particular M.C.
is grateful to Roberta Merlo for sharing her deep knowledge about spheres with
handles. We are particular indebted with Manoj Mandal and Federico Gasparotto
for providing us with the example of Sec. 8, including all the detailed
calculations shown there.

##  Appendix A Baikov Representation

For the sake of completeness, in this Appendix we provide a short derivation
of the Baikov formula (2). The idea is to rewrite the integral using the
independent scalar products between momenta as integration variables. They are

$M=LE+L(L+1)/2\,,$ (A.1)

where the first term is the number of scalar products $q_{i}\cdot p_{j}$
between one loop and one external momentum, while the second represents the
scalar products $q_{i}\cdot q_{j}$ between loop momenta.
In total, one has $m=E+L$ total independent momenta. It is useful to introduce
the complete vector $k$ of all $m$ momenta:

$k=(\underbrace{q_{1},\cdots q_{L}}_{K_{i}=q_{i}\,\,\,i\leq
L}\,\,\,\,,\,\,\,\underbrace{p_{1},\cdots p_{E}}_{K_{i}=p_{i-L}\,\,\,i>L}).$
(A.2)

To perform this change of variables, we start by decomposing $q_{1}$ as

$\displaystyle q_{1}=q_{1\parallel}+q_{1\perp}\,,$ (A.3)

where $q_{1\parallel}$ represents the projection of $q_{1}$ onto the space
generated by all the other $m-1=E+L-1$ momenta $\left\\{q_{2},\cdots
q_{L},p_{1},\cdots,p_{E}\right\\}$, while $q_{1\perp}$ represents the
orthogonal component with respect to said space. This is done for every
variable: for each $i$, the corresponding $q_{i}$ is projected onto the space
generated by the momenta which come next in the vector $k$:
$\left\\{q_{i+1},\cdots q_{L},p_{1},\cdots,p_{E}\right\\}$. In the last step,
one projects the last $q_{L}$ along the space of the $E$ external momenta
$\left\\{p_{1},\cdots,p_{E}\right\\}$. This decomposition leads to

$d^{D}q_{1}\cdots
d^{D}q_{L}=(d^{E+L-1}q_{1\parallel}\,\,d^{D-E-L+1}q_{1\perp})\,\,\cdots\,\,(d^{E}q_{L\parallel}\,\,d^{D-E}q_{L\perp})\,.$
(A.4)

We introduce the Gram matrix

$G(k)=\begin{pmatrix}s_{11}\,\,\,\cdots\,\,\,s_{1m}\\\
\vdots\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\vdots\\\
s_{m1}\,\,\,\cdots\,\,\,s_{mm}\end{pmatrix}\,,$ (A.5)

in which the entries are $s_{ij}=k_{i}k_{j}$. The square root of the
determinant (which we will adress as $G$ instead of $\det(G)$ for brevity) of
the matrix (A.5) represents the volume of the parallelotope generated by the
elements of $k$. With this interpretation, one can write

  * 1:

Parallel component $q_{i\parallel}$.

$\displaystyle d^{E+L-i}q_{i\parallel}=\frac{ds_{i,i+1}\cdot ds_{i,i+2}\cdots
ds_{i,E+L}}{G^{1/2}(q_{i+1},\cdots,q_{L},p_{1},\cdots,p_{E})}=\prod_{j=i+1}^{E+L}\frac{ds_{ij}}{G^{1/2}(q_{i+1},\cdots,q_{L},p_{1},\cdots,p_{E})}\,.$
(A.6)

In the numerator of Eq. (A.6) we perform the scalar product of $q_{i}$ along
the space generated by the vectors which come next (starting from $q_{i+1}$):
this allows to find the projections of $q_{i}$ along such vectors. The
denominator is the necessary normalization which allows to get the correct
dimension.

  * 2:

Perpendicular component $q_{i\perp}$.
Introducing polar coordinates and separating the angular part from the radial
part, we get

$\displaystyle d^{D-E-L+i}q_{i\perp}$
$\displaystyle=\Omega_{D-E-L+i-1}|q_{i\perp}|^{D-E-L+i-1}d|q_{i\perp}|$ (A.7)
$\displaystyle=\frac{\Omega_{D-E-L+i-1}}{2}|q_{i\perp}|^{D-E-L+i-2}d|q_{i\perp}|^{2}\,,$
(A.8)

where $d|q_{i\perp}|^{2}$ is $ds_{ii}$. Notice that $|q_{i\perp}|$ can be seen
as the height of a parallelotope with base $q_{i+1},\cdots,p_{E}$, so it can
be computed as the whole volume divided by the volume of its base: hence, we
write

$d^{D-E-L+i}q_{i\perp}=\frac{\Omega_{D-E-L+i-1}}{2}\left(\frac{G(q_{i},\cdots
q_{L},p_{1},\cdots,p_{E})}{G(q_{i+1},\cdots
q_{L},p_{1},\cdots,p_{E})}\right)^{\frac{D-E-L-2+i}{2}}ds_{ii}\,.$ (A.9)

Using Eq. (A.6) and Eq. (A.9) and recalling

$\frac{\Omega_{D-E-L+i-1}}{2}=\frac{\pi^{\frac{D-E-L+i}{2}}}{\Gamma(\frac{D-E-L+i}{2})}\,,$
(A.10)

it is possible to obtain

$\displaystyle\frac{d^{D}q_{i}}{\pi^{D}/2}$
$\displaystyle=\frac{\pi^{\frac{-E-L+i}{2}}}{\Gamma(\frac{D-E-L+i}{2})}\int_{\Gamma_{i}}G^{-1/2}(q_{i+1},\cdots,p_{E})\prod_{j=i}^{E+L}d(q_{i}k_{j})\left(\frac{G(q_{i},\cdots,p_{E})}{G(q_{i+1},\cdots
p_{E})}\right)^{\frac{D-E-L-2+i}{2}}\,,$ (A.11)

where $\Gamma_{i}$ is the contour determined by $|q_{i\perp}|^{2}>0$. The
whole Feynman integral (1) becomes

$\displaystyle I_{\nu_{1},\cdots,\nu_{N}}$
$\displaystyle=\frac{\pi^{\frac{L-M}{2}}}{\prod_{l=1}^{L}\Gamma(\frac{D-E-L+l}{2})}\int_{\Gamma}\prod_{i=1}^{L}G^{-1/2}(q_{i+1},\cdots,p_{E})\left(\frac{G(q_{i},\cdots,p_{E})}{G(q_{i+1},\cdots
p_{E})}\right)^{\frac{D-E-L-2+i}{2}}\times$ (A.12)
$\displaystyle\times\frac{\prod_{j=i}^{E+L}d(q_{i}k_{j})}{\prod_{a=1}^{N}D_{a}^{\nu_{a}}}\,.$
(A.13)

By explicit computation of the product over $i$ in Eq. (A.13), only a few
terms survive and Eq. (A.13) becomes

$\displaystyle I_{\nu_{1},\cdots,\nu_{N}}$
$\displaystyle=\frac{\pi^{\frac{L-M}{2}}G(p_{1},\cdots,p_{E})^{-\frac{L}{2}}}{\prod_{i=1}^{L}\Gamma(\frac{D-E-L+i}{2})}\int_{\Gamma}\left(\frac{G(q_{1},\cdots,p_{E})}{G(p_{1},\cdots
p_{E})}\right)^{\frac{D-E-L-1}{2}}\frac{\prod_{i=1}^{L}\prod_{j=1}^{E+L}d(q_{i}k_{j})}{\prod_{a=1}^{N}D_{a}^{\nu_{a}}}$
$\displaystyle\equiv
C\int_{\Gamma}\textsl{B}^{\gamma}\frac{\prod_{i=1}^{L}\prod_{j=1}^{E+L}d(q_{i}k_{j})}{\prod_{a=1}^{N}D_{a}^{\nu_{a}}}\,.$
(A.14)

In the second line of Eq. (A.14), B is the Baikov polynomial (similar to the
Jacobian associated to the change of variables) and is raised to the power
$\gamma=(D-E-L-1)/2$, while $C$ is an overall constant. Eq. (A.14) can be
further simplified by noticing that in general the propagators $D_{a}$ in the
denominator usually depend linearly in the scalar products between momenta,
hence one can perform a change of variables $z_{a}=D_{a}$. Notice however that
the number of denominators $N$ and the the number of independent momenta
scalar products $M$ are different in general: for this reason it is sufficient
to add $N-M$ fake denominators $D_{a}$, each one raised to a certain power
$\nu_{a}$. The original integral can be recovered by putting $\nu_{a}=0$ for
$a=N+1,\cdots,M$. With this procedure one readily obtains the final form (2),
where the constant $K=C/\det A$, and $A$ is the Jacobian matrix between the
scalar products and the propagators $D_{a}=A_{a}^{ij}q_{i}k_{j}+m_{a}^{2}$.

##  Appendix B An introduction on Gröbner bases

In this Appendix we introduce some basic tools, along with some notations and
definitions, necessary to understand how to work with Gröbner bases. We start
by calling $R=\mathbb{K}(z_{1},\cdots,z_{n})$ the ring over a field
$\mathbb{K}$ with $n$ variables $z_{1},\cdots,z_{n}$, with $I\subset R$ being
an ideal in the ring $R$.

######  Definition 2.

Combination of polynomials
Given a set of polynomials, an expression consisting of the sum of polynomial
multiples of the elements in the set is called a combination.

For example,
$(3z_{1}+{z_{2}}^{2})(z_{1}+z_{3})+z_{1}({z_{1}}^{2}+5z_{2}^{2}z_{3}^{2})$ is
a combination of the polynomials
$\\{z_{1}+z_{3},z_{1}^{2}+5z_{2}^{2}z_{3}^{2}\\}$.

######  Definition 3.

Power product
A power product is a polynomial which can be obtained only by multiplication
of the variables.

For example, $z_{1}z_{2}^{2}$ is a power product.

######  Definition 4.

Term order
A term order on the monomials of a ring $R$ is an order $\prec$ with the
following properties:

  * 1:

$M\prec N\Leftrightarrow MP\preccurlyeq NP$

  * 2:

$M\preccurlyeq MP$

For example, the following order is called a Lexicographic order on the
variables $z_{1},z_{2}$:

$1\prec z_{1}\prec z_{1}^{2}\prec z_{1}^{3}\prec\cdots\prec z_{2}\prec
z_{1}z_{2}\prec z_{1}^{2}z_{2}\prec\cdots z_{2}^{2}\prec z_{1}z_{2}^{2}\prec
z_{1}^{2}z_{2}^{2}\prec\cdots$ (B.1)

######  Definition 5.

Initial term (or leading power product)
Given a ring $R$ with a term order $\prec$, the initial term $in_{\succ}(f)$
of a polynomial $f\in R$ (or leading power product $LLP_{\succ}(f))$ is the
biggest monomial in $f$ with respect to the given order, together with its
coefficient.

For example, if $z_{1}\prec z_{2}$ and $f=3z_{1}z_{2}+2z_{2}^{2}$, then
$in_{\succ}(f)=2z_{2}^{2}$.

######  Definition 6.

Initial Ideal
Given an ideal $I\subset R$, $in_{\succ}(I)$ is the monomial ideal generated
by the initial terms $\left\\{in_{\succ}(f):f\in I\right\\}$.

######  Definition 7.

Gröbner basis
Given an ideal $I\subset R$ with a term order $\prec$ and a set of polynomials
$G=\left\\{g_{1},\cdots,g_{s}\right\\}\subset I$, if $in_{\succ}(I)$ is
generated by $\left\\{in_{\succ}(g_{1}),\cdots,in_{\succ}(g_{s})\right\\}$
then $G$ is called a Gröbner basis with respect to the order.

Definition 7 implies that, given a $f\in I$ with $f\neq 0$, $in_{\succ}(f)$
must be divisible by at least one of the $in_{\succ}(g_{i})$ in the basis.

We also introduce some concepts which will be useful in our context. First, we
introduce the algorithm of division of a polynomial by a given set of
polynomials. This concept arises from the following question: given an ideal
$I\subset R$, when is a $h\in R$ also $h\in I$? Suppose we have a Gröbner
basis for $I$: if $h\in I$, then its leading power product must be divisible
by one of the leading power products appearing in the basis (the one belonging
to a certain polynomial $g_{i}$, to fix ideas). The subtraction of an
appropriate multiple of $g_{i}$ from $h$ leads to a new $\tilde{h}$, where the
old leading term has been cancelled. If $h\in I$, then also $\tilde{h}\in I$:
the process can be reiterated until $0$ is obtained (hence $h\in I$ is
verified). If the iterated process leads to a polynomial $\neq 0$ which cannot
be further reduced, then $h\notin I$. The result of this procedure of division
of a polynomial $f$ by a set of polynomials $G$ is called the remainder
$R_{G}(f)$. Notice that this procedure must be finite, as the definition of a
term order $\prec$ implies a descending path in the variables which has to end
(this is not true for a generic set of polynomials which is not a Gröbner
basis for $I$, as one should check infinite types of combinations of
polynomials in principle).
Notice that the concept of remainder can be used to determine if two
polynomials $f,g$ in $R$ are equal modulo $I$ (i.e. $f\sim g$ if $f+$
combination of generators of $I=g)$: in general one should check infinite
combinations of the generators, but with a Gröbner basis for $I$:

  * •

$G$ is a Gröbner basis for $I\Leftrightarrow R_{G}(f)=0\,\,\forall\,\,f\in I$;

  * •

If $G$ is a Gröbner basis for $I$, then $f,g\in R$ are equal modulo
$I\Leftrightarrow R_{G}(f)=R_{G}(g)$.

With these tools, it is possible to construct a practical way which tells us
how to build Gröbner bases.

######  Definition 8.

S-polynomial
Given two polynomials $f,g$, the S-polynomial between $f$ and $g$ is defined
as:

$S(f,g)=\text{lcm}(\tilde{in}_{\succ}(f),\tilde{in}_{\succ}(g))\left(\frac{f}{in_{\succ}(f)}-\frac{g}{in_{\succ}(g)}\right)\,,$
(B.2)

where the notation $\tilde{in}_{\succ}(f)$ means $in_{\succ}(f)$ deprived of
its coefficient.

Notice that the S-polynomial of two polynomials $f,g$ is a combination of $f$
and $g$: because of the previous observations, if a set
$G=\left\\{g_{1},\cdots,g_{s}\right\\}$ is Gröbner basis, then $\forall\,i\neq
j$ $R_{G}(S(g_{i},g_{j}))=0$. This requirement is the fundament of
Buchberger’s Algorithm [145], which, starting from a certain set of
polynomials $\left\\{f_{1},\cdots,f_{t}\right\\}$ which generate an ideal $I$,
returns a Gröbner basis for the same ideal, fixed a certain order. The
algorithm works as follows: given $F=\left\\{f_{1},\cdots,f_{t}\right\\}$, it
computes $S(f_{1},f_{2})$; if $R_{F}(S(f_{1},f_{2}))=0$, it proceeds to the
next couple of elements ($f_{1}$ and $f_{3}$, for instance); if
$R_{F}(S(f_{1},f_{2}))\neq 0$, $F$ is not a Gröbner basis: it then adds
$S(f_{1},f_{2})$ \- or better yet, its remainder $R_{F}(S(f_{1},f_{2}))$ \- as
a new element of $F$ and starts again. Notice that the addition of the new
term to $F$ ensures that in the next iteration $R_{F}(S(f_{1},f_{2})=0$. At
the end of the process, $F$ forms a Gröbner basis.

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