Computing Regular Meromorphic Differential Forms via Saito’s Logarithmic
Residues

Computing Regular Meromorphic Differential Forms
via Saito’s Logarithmic Residues††This paper is a contribution to the Special
Issue on Primitive Forms and Related Topics in honor of Kyoji Saito for his
77th birthday. The full collection is available at
https://www.emis.de/journals/SIGMA/Saito.html

Shinichi TAJIMA a and Katsusuke NABESHIMA b

S. Tajima and K. Nabeshima

a) Graduate School of Science and Technology, Niigata University,
a) 8050, Ikarashi 2-no-cho, Nishi-ku Niigata, Japan
<EMAIL_ADDRESS>
b) Graduate School of Technology, Industrial and Social Sciences, Tokushima
University,
b) 2-1, Minamijosanjima-cho, Tokushima, Japan<EMAIL_ADDRESS>
Received July 24, 2020, in final form February 05, 2021; Published online
February 27, 2021

Logarithmic differential forms and logarithmic vector fields associated to a
hypersurface with an isolated singularity are considered in the context of
computational complex analysis. As applications, based on the concept of
torsion differential forms due to A.G. Aleksandrov, regular meromorphic
differential forms introduced by D. Barlet and M. Kersken, and Brieskorn
formulae on Gauss–Manin connections are investigated. $(i)$ A method is given
to describe singular parts of regular meromorphic differential forms in terms
of non-trivial logarithmic vector fields via Saito’s logarithmic residues. The
resulting algorithm is illustrated by using examples. $(ii)$ A new link
between Brieskorn formulae and logarithmic vector fields is discovered and an
expression that rewrites Brieskorn formulae in terms of non-trivial
logarithmic vector fields is presented. A new effective method is described to
compute non trivial logarithmic vector fields which are suitable for the
computation of Gauss–Manin connections. Some examples are given for
illustration.

logarithmic vector field; logarithmic residue; torsion module; local
cohomology

32S05; 32A27

Dedicated to Kyoji Saito

on the occasion of his $77^{th}$ birthday

##  1 Introduction

In 1975, K. Saito introduced, with deep insight, the concept of logarithmic
differential forms and that of logarithmic vector fields and studied
Gauss–Manin connection associated with the versal deformations of hypersurface
singularities of type $A_{2}$ and $A_{3}$ as applications. These results were
published in [33]. He developed the theory of logarithmic differential forms,
logarithmic vector fields and the theory of residues and published in 1980 a
landmark paper [34]. One of the motivations of his study, as he himself wrote
in [34], came from the study of Gauss–Manin connections [5, 32]. Another
motivation came from the importance of these concepts he realized. Notably the
logarithmic residue, interpreted as a meromorphic differential form on a
divisor, is regarded as a natural generalization of the classical Poincaré
residue to the singular cases.

In 1990, A.G. Aleksandrov [2] studied Saito theory and gave in particular a
characterization of the image of the residue map. He showed that the image
sheaf of the logarithmic residues coincides with the sheaf of regular
meromorphic differential forms introduced by D. Barlet [5] and M. Kersken [15,
16]. We refer the reader to [4, 8, 9, 10, 12, 29, 30] for more recent results
on logarithmic residues.

We consider logarithmic differential forms along a hypersurface with an
isolated singularity in the context of computational complex analysis. In our
previous paper [40], we study torsion modules and give an effective method for
computing them. In the present paper, we first consider a method for computing
regular meromorphic differential forms. We show that, based on the result of
A.G. Aleksandrov mentioned above, representatives of regular meromorphic
differential forms can be computed by adapting the method presented in [40] on
torsion modules. Main ideas of our approach are the use of the concept of
logarithmic residues and that of logarithmic vector fields. Next, we discuss a
relation between logarithmic differential forms and Brieskorn formulae [5, 35,
37] and we show that Brieskorn formulae can be rewritten in terms of
logarithmic vector fields. Applications to the computation of Gauss–Manin
connections are illustrated by using examples.

In Section 2, we briefly recall some basics on logarithmic differential forms,
logarithmic residues, Barlet sheaf and torsion differential forms. In Section
3, we first recall the notion of logarithmic vector fields and a result gave
in [40] to show that torsion differential forms can be described in terms of
non trivial logarithmic vector fields. Next, we recall our previous results to
show that non-trivial logarithmic vector fields can be computed by using a
polar method and local cohomology. Lastly in Section 3, we present Theorem
3.11 which say that regular meromorphic differential forms can be explicitly
computed by modifying our previous algorithm on torsion differential forms. In
Section 4, we give some examples to illustrate the proposed method of
computing non-trivial logarithmic vector fields and regular meromorphic
differential forms. In Section 5, we consider Brieskorn formulae on
Gauss–Manin connections. We show that Brieskorn formulae described in terms of
logarithmic differential forms can be rewritten in terms of non-trivial
logarithmic vector fields. We give a new method for computing non-trivial
logarithmic vector fields which is suitable in use to compute a connection
matrix of Gauss–Manin connections. Finally, we show that the use of integral
dependence relations provides a new effective tool for computing saturations
of Gauss–Manin connection.

##  2 Logarithmic differential forms and residues

In this section, we briefly recall the concept of logarithmic differential
forms and that of logarithmic residues and fix notation. We refer the reader
to [34] for details. Next we recall the result of A.G. Aleksandrov on regular
meromorphic differential forms. Then, we recall a result of G.-M. Greuel on
torsion modules.

Let $X$ be an open neighborhood of the origin $O$ in ${\mathbb{C}}^{n}$. Let
${\mathcal{O}}_{X}$ be the sheaf on $X$ of holomorphic functions and
${\mathcal{O}}_{X,O}$ the stalk at $O$ of the sheaf ${\mathcal{O}}_{X}$.

###  2.1 Logarithmic residues

Let $f$ be a holomorphic function defined on $X$. Let $S=\\{x\in
X\,|\,f(x)=0\\}$ denote the hypersurface defined by $f$.

######  Definition 2.1.

Let $\omega$ be a meromorphic differential $q$-form on $X$, which may have
poles only along $S$. The form $\omega$ is a logarithmic differential form
along $S$ if it satisfies the following equivalent four conditions:

  1. $(i)$

$f\omega$ and $f{\rm d}\omega$ are holomorphic on $X$.

  2. $(ii)$

$f\omega$ and ${\rm d}f\wedge\omega$ are holomorphic on $X$.

  3. $(iii)$

There exist a holomorphic function $g(x)$ and a holomorphic $(q-1)$-form $\xi$
and a holomorphic $q$-form $\eta$ on $X$, such that:

    1. $(a)$

$\dim_{\mathbb{C}}(S\cap\\{x\in X\,|\,g(x)=0\\})\leq n-2,$

    2. $(b)$

${\displaystyle g\omega=\frac{{\rm d}f}{f}\wedge\xi+\eta.}$

  4. $(iv)$

There exists an $(n-2)$-dimensional analytic set $A\subset S$ such that the
germ of $\omega$ at any point $p\in S-A$ belongs to ${\frac{{\rm
d}f}{f}\wedge\Omega_{X,p}^{q-1}+\Omega_{X,p}^{q}},$ where $\Omega_{X,p}^{q}$
denotes the module of germs of holomorphic $q$-forms on $X$ at $p$.

For the equivalence of the condition above, see [34]. Let $\Omega_{X}^{q}(\log
S)$ denote the sheaf of logarithmic $q$-forms along $S$. Let
${\mathcal{M}}_{S}$ be the sheaf on $S$ of meromorphic functions, let
$\Omega_{S}^{q}$ be the sheaf on $S$ of holomorphic $q$-forms defined to be

$\displaystyle\Omega_{S}^{q}=\Omega_{X}^{q}/\big{(}f\Omega_{X}^{q}+{\rm
d}f\wedge\Omega_{X}^{q-1}\big{)}.$

######  Definition 2.2.

The residue map ${\rm res}\colon\Omega_{X}^{q}(\log
S)\longrightarrow{\mathcal{M}}_{S}\otimes_{{\mathcal{O}}_{X}}\Omega_{S}^{q-1}$
is defined as follows: For $\omega\in\Omega_{S}^{q}(\log S)$, by definition,
there exist $g$, $\xi$ and $\eta$ such that

  1. $(a)$

$\dim_{\mathbb{C}}(S\cap\\{x\in X\,|\,g(x)=0\\})\leq n-2$, and

  2. $(b)$

${\displaystyle g\omega=\frac{{\rm d}f}{f}\wedge\xi+\eta}$.

Then the residue of $\omega$ is defined to be ${\rm
res}(\omega)=\frac{\xi}{g}\Big{|}_{S}$ in
${\mathcal{M}}_{S}\otimes_{{\mathcal{O}}_{X}}\Omega_{S}^{q-1}$.

Note that it is easy to see that the image sheaf of the residue map ${\rm
res}$ of the subsheaf ${\frac{{\rm
d}f}{f}\wedge\Omega_{X}^{q-1}+\Omega_{X}^{q}}$ of $\Omega_{X}^{q}(\log S)$ is
equal to
$\left.\Omega_{X}^{q-1}\middle|\raisebox{-3.01385pt}[4.30554pt][-8.61108pt]{{\;${}_{S}$}}\right.$:

$\displaystyle{\rm res}\bigg{(}\frac{{\rm
d}f}{f}\wedge\Omega_{X}^{q-1}+\Omega_{X}^{q}\bigg{)}=\left.\Omega_{X}^{q-1}\middle|\raisebox{-3.01385pt}[4.30554pt][-8.61108pt]{{\;${}_{S}$}}\right..$

See also [34] for details on logarithmic residues. The concept of residues for
logarithmic differential forms can be actually regarded as a natural
generalization of the classical Poincaré residue.

###  2.2 Barlet sheaf and torsion differential forms

In 1978, by using results of F. El Zein on fundamental classes, D. Barlet
introduced in [5] the notion of the sheaf $\omega_{S}^{q}$ of regular
meromorphic differential forms in a quite general setting. He showed that for
the case $q=n-1$, the sheaf $\omega_{S}^{n-1}$ coincides with the Grothendieck
dualizing sheaf and $\omega_{S}^{q}$ can also be defined in the following
manner.

######  Definition 2.3.

Let $S$ be a hypersurface in $X\subset{\mathbb{C}}^{n}$. Let
$\omega_{S}^{n-1}$ be the Grothendieck dualizing sheaf ${\rm
Ext}_{{\mathcal{O}}_{X}}^{1}\big{(}{\mathcal{O}}_{S},\Omega_{X}^{n}\big{)}$.
Then, the sheaf of regular meromorphic differential forms $\omega_{S}^{q}$,
$q=0,1,\dots,n-2$ on $S$ is defined to be

$\displaystyle\omega_{S}^{q}={\rm
Hom}_{{\mathcal{O}}_{S}}\big{(}\Omega_{S}^{n-1-q},\omega_{S}^{n-1}\big{)}.$

In 1990, A.G. Aleksandrov [2] obtained the following result.

######  Theorem 2.4.

For any $q\geq 0$, there is an isomorphism of ${\mathcal{O}}_{S}$ modules

$\displaystyle{\rm res}\big{(}\Omega_{X}^{q}(\log
S)\big{)}\cong\omega_{S}^{q-1}.$

See [2] or [3] for the proof.

Let ${\rm Tor}(\Omega_{S}^{q})$ denote the sheaf of torsion differential
$q$-forms of $\Omega_{S}^{q}$.

######  Example 2.5.

Let $X$ be an open neighborhood of the origin $O$ in $\mathbb{C}^{2}.$ Let
$f(x,y)=x^{2}-y^{3}$ and $S=\\{(x,y)\in X\,|\,f(x,y)=0\\}$. Then, for stalk at
the origin of the sheaves of logarithmic differential forms, we have

$\displaystyle\Omega_{X,O}^{1}(\log
S)\cong{\mathcal{O}}_{X,O}\bigg{(}\frac{{\rm
d}f}{f},\frac{\beta}{f}\bigg{)},\qquad\Omega_{X,O}^{2}(\log
S)\cong{\mathcal{O}}_{X,O}\bigg{(}\frac{{\rm d}x\wedge{\rm d}y}{f}\bigg{)},$

where ${\mathcal{O}}_{X,O}$ is the stalk at the origin of the sheaf
${\mathcal{O}}_{X}$ of holomorphic functions and $\beta=2y{\rm d}x-3x{\rm
d}y$. The differential form $\beta$, as an element of
$\Omega_{S}^{1}=\Omega_{X}^{1}/\big{(}\mathcal{O}_{X}{\rm
d}f+f\Omega_{X}^{1}\big{)}$, is a torsion. The differential form $y\beta$ is
also a torsion. Since the defining function $f$ is quasi-homogeneous, the
dimension of the vector space ${\rm Tor}\big{(}\Omega_{S}^{1}\big{)}$ is equal
to the Milnor number $\mu=2$ of $S$ [18, 47]. Therefore we have ${\rm
Tor}\big{(}\Omega_{S}^{1}\big{)}\cong{\mathcal{O}}_{X,O}(\beta)\cong{\mathbb{C}}(\beta,y\beta)$.

In 1988 [1], A.G. Aleksandrov studied logarithmic differential forms and
residues and proved in particular the following.

######  Theorem 2.6.

Let $S=\\{x\in X\,|\,f(x)=0\\}$ be a hypersurface in
$X\subset{\mathbb{C}}^{n}$. For $q=0,1,\ldots,n$, there exists an exact
sequence of sheaves of ${\mathcal{O}}_{X}$ modules,

$\displaystyle 0\longrightarrow\frac{{\rm
d}f}{f}\wedge\Omega_{X}^{q-1}+\Omega_{X}^{q}\longrightarrow\Omega_{X}^{q}(\log
S)\stackrel{{\scriptstyle\cdot f}}{{\longrightarrow}}{\rm
Tor}\big{(}\Omega_{S}^{q}\big{)}\longrightarrow 0.$

The result above yields the following observation: ${\rm
Tor}\big{(}\Omega_{S}^{q}\big{)}$ plays a key role to study the structure of
${\rm res}\big{(}\Omega_{X}^{q}(\log S)\big{)}$.

###  2.3 Vanishing theorem

In 1975, in his study [13] on Gauss–Manin connections G.-M. Greuel proved the
following results on torsion differential forms.

######  Theorem 2.7.

Let $S=\\{x\in X\,|\,f(x)=0\\}$ be a hypersurface in $X$ with an isolated
singularity at $O\in{\mathbb{C}}^{n}$. Then,

  1. $(i)$

${\rm Tor}\big{(}\Omega_{S}^{q}\big{)}=0$, $q=0,1,\ldots,n-2$.

  2. $(ii)$

${\rm Tor}\big{(}\Omega_{S}^{n-1}\big{)}$ is a skyscraper sheaf supported at
the origin $O$.

  3. $(iii)$

The dimension, as a vector space over ${\mathbb{C}}$, of the torsion module
${\rm Tor}\big{(}\Omega_{S}^{n-1}\big{)}$ is equal to $\tau(f)$, the Tjurina
number of the hypersurface $S$ at the origin defined to be

$\displaystyle\tau(f)=\dim_{{\mathbb{C}}}\bigg{(}{\mathcal{O}}_{X,O}\Big{/}\bigg{(}f,\frac{\partial
f}{\partial x_{1}},\frac{\partial f}{\partial x_{2}},\ldots,\frac{\partial
f}{\partial x_{n}}\bigg{)}\bigg{)},$

where $\big{(}f,\frac{\partial f}{\partial x_{1}},\frac{\partial f}{\partial
x_{2}},\ldots,\frac{\partial f}{\partial x_{n}}\big{)}$ is the ideal in
${\mathcal{O}}_{X,O}$ generated by $f,\frac{\partial f}{\partial
x_{1}},\frac{\partial f}{\partial x_{2}},\ldots,\frac{\partial f}{\partial
x_{n}}$.

Note that the first result was obtained by U. Vetter in [46] and the last
result above is a generalization of a result of O. Zariski [47]. G.-M. Greuel
obtained much more general results on torsion modules. See [13, Proposition
1.11, p. 242].

Assume that the hypersurface $S$ has an isolated singularity at the origin. We
thus have, by combining the results of G.-M. Greuel above and of A.G.
Aleksandrov presented in the previous section, the following:

  1. $(i)$

$\Omega_{X,O}^{q}(\log S)=\frac{{\rm
d}f}{f}\wedge\Omega_{X,O}^{q-1}+\Omega_{X,O}^{q}$, $q=1,2,\ldots,n-2$,

  2. $(ii)$

$0\longrightarrow\frac{{\rm
d}f}{f}\wedge\Omega_{X,O}^{n-2}+\Omega_{X,O}^{n-1}\longrightarrow\Omega_{X,O}^{n-1}(\log
S)\stackrel{{\scriptstyle\cdot f}}{{\longrightarrow}}{\rm
Tor}\big{(}\Omega_{S}^{n-1}\big{)}\longrightarrow 0$.

Accordingly we have the following.

######  Proposition 2.8.

Let $S=\\{x\in X\,|\,f(x)=0\\}$ be a hypersurface in $X$ with an isolated
singularity at $O\in{\mathbb{C}}^{n}$. Then, $\omega_{S}^{q}=\Omega_{X}^{q}$,
$q=0,1,\ldots,n-3$ holds.

###### Proof.

Since $\left.{\rm res}\big{(}\Omega_{X}^{q}(\log
S)\big{)}=\Omega_{X}^{q-1}\middle|\raisebox{-3.01385pt}[4.30554pt][-8.61108pt]{{\;${}_{S}$}}\right.$,
$q=1,2,\ldots,n-2$, the result of A.G. Aleksandrov presented in the last
section yields the result. ∎

##  3 Description via logarithmic residues

In this section, we recall results given in [40] to show that torsion
differential forms can be described in terms of non-trivial logarithmic vector
fields. We also recall basic ideas and the framework for computing non-trivial
logarithmic vector fields. As an application, we give a method for computing
logarithmic residues.

###  3.1 Logarithmic vector fields

A vector field $v$ on $X$ with holomorphic coefficients is called logarithmic
along the hypersurface $S$, if the holomorphic function $v(f)$ is in the ideal
$(f)$ generated by $f$ in ${\mathcal{O}}_{X}$. Let ${\mathcal{D}}{\rm
er}_{X}(-\log S)$ denote the sheaf of modules on $X$ of logarithmic vector
fields along $S$ [34].

Let $\omega_{X}={\rm d}x_{1}\wedge{\rm d}x_{2}\wedge\cdots\wedge{\rm d}x_{n}$.
For a holomorphic vector field $v$, let $i_{v}(\omega_{X})$ denote the inner
product of $\omega_{X}$ by $v$.

######  Proposition 3.1.

Let $S=\\{x\in X\,|\,f(x)=0\\}$ be a hypersurface with an isolated singularity
at the origin. Then, $\Omega_{X,O}^{n-1}(\log S)$ is isomorphic to
${\mathcal{D}}{\rm er}_{X,O}(-\log S)$, more precisely

$\displaystyle\Omega_{X,O}^{n-1}(\log
S)=\left\\{\frac{i_{v}(\omega_{X})}{f}\middle|\ v\in{\mathcal{D}}{\rm
er}_{X,O}(-\log S)\right\\}$

holds.

###### Proof.

Let $\beta=i_{v}(\omega_{X}),$ and set $\omega=\frac{\beta}{f}$. Then,
$f\omega=\beta$ is a holomorphic differential form. Therefore, the meromorphic
differential $n-1$ form $\omega$ is logarithmic if and only if ${\rm
d}f\wedge\frac{\beta}{f}$ is a holomorphic differential $n$-form. Since ${\rm
d}f\wedge\beta={\rm d}f\wedge i_{v}(\omega_{X})=v(f)\omega_{X}$, we have ${\rm
d}f\wedge\frac{\beta}{f}=\frac{v(f)}{f}\omega_{X}$. Hence, the condition above
means $v(f)$ is in the ideal $(f)\subset{\mathcal{O}}_{X,O}$ generated by $f$.
This completes the proof. ∎

A germ of logarithmic vector field $v$ generated over ${\mathcal{O}}_{X,O}$ by

$\displaystyle f\frac{\partial}{\partial x_{i}},\qquad
i=1,2,\ldots,n,\qquad\frac{\partial f}{\partial x_{j}}\frac{\partial}{\partial
x_{i}}-\frac{\partial f}{\partial x_{i}}\frac{\partial}{\partial x_{j}},\qquad
1\leq i<j\leq n,$

is called trivial.

######  Lemma 3.2.

Let $v$ be a germ of a logarithmic vector field. Then, the following
conditions are equivalent:

  1. $(i)$

$\displaystyle\omega=\frac{i_{v}(\omega_{X})}{f}$ belongs to
$\displaystyle\frac{{\rm d}f}{f}\wedge\Omega_{X,O}^{n-2}+\Omega_{X,O}^{n-1}$,

  2. $(ii)$

$v$ is a trivial vector field.

###### Proof.

The logarithmic differential form $\omega=\frac{i_{v}(\omega_{X})}{f}$ is in
$\Omega_{X,O}^{n-1}+\frac{{\rm d}f}{f}\wedge\Omega_{X,O}^{n-2}$ if and only if
the numerator $i_{v}(\omega_{X})$ is in $f\Omega_{X,O}^{n-1}+{\rm
d}f\wedge\Omega_{X,O}^{n-2}$. The last condition is equivalent to the
triviality of the vector field $v$, which completes the proof. ∎

For $\beta\in\Omega_{X,O}^{n-1}$, let $[\beta]$ denote the Kähler differential
form in $\Omega_{S,O}^{n-1}$ defined by $\beta$, that is, $[\beta]$ is the
equivalence class in $\Omega_{X,O}^{n-1}/\big{(}f\Omega_{X,O}^{n-1}+{\rm
d}f\wedge\Omega_{X,O}^{n-2}\big{)}$ of $\beta$.

The lemma above amount to say that, for logarithmic vector fields $v$,
$[i_{v}(\omega_{X})]$ is a non-zero torsion differential form in ${\rm
Tor}\big{(}\Omega_{S,O}^{n-1}\big{)}$ if and only if $v$ is a non-trivial
logarithmic vector field.

We say that germs of two logarithmic vector fields
$v,v^{\prime}\in{\mathcal{D}}{\rm er}_{X,O}(-\log S)$ are equivalent, denoted
by $v\sim v^{\prime}$, if $v-v^{\prime}$ is trivial. Let ${\mathcal{D}}{\rm
er}_{X,O}(-\log S)/{\sim}$ denote the quotient by the equivalence relation
$\sim$. (See [39].)

Now consider the following map

$\displaystyle\Theta\colon\ {\mathcal{D}}{\rm er}_{X,O}(-\log
S)/{\sim}\longrightarrow\Omega_{X,O}^{n-1}/\big{(}f\Omega_{X,O}^{n-1}+{\rm
d}f\wedge\Omega_{X,O}^{n-2}\big{)}$

defined to be $\Theta([v])=[i_{v}(\omega_{X})],$ where $[v]$ is the
equivalence class in ${\mathcal{D}}{\rm er}_{X,O}(-\log S)/{\sim}$ of $v$. It
is easy to see that the map $\Theta$ is well-defined. We arrive at the
following description of the torsion module.

######  Theorem 3.3 ([40]).

The map

$\displaystyle\Theta\colon\ {\mathcal{D}}{\rm er}_{X,O}(-\log
S)/{\sim}\longrightarrow{\rm Tor}\big{(}\Omega_{S}^{n-1}\big{)}$

is an isomorphism.

###  3.2 Polar method

In [39], based on the concept of polar variety, logarithmic vector fields are
studied and an effective and constructive method is considered. Here in this
section, following [27, 39] we recall some basics and give a description of
non-trivial logarithmic vector fields.

Let $S=\\{x\in X\,|\,f(x)=0\\}$ be a hypersurface with an isolated
singularity. In what follows, we assume that $f,\frac{\partial f}{\partial
x_{2}},\frac{\partial f}{\partial x_{3}},\ldots,\frac{\partial f}{\partial
x_{n}}$ is a regular sequence and the common locus $V\big{(}f,\frac{\partial
f}{\partial x_{2}},\frac{\partial f}{\partial
x_{3}},\ldots,\allowbreak\frac{\partial f}{\partial x_{n}}\big{)}\cap X$ is
the origin $O$. See [19] for an algorithm of testing zero-dimensionality of
varieties at a point.

Let $\big{(}f,\frac{\partial f}{\partial x_{2}},$ $\frac{\partial f}{\partial
x_{3}},\ldots,\frac{\partial f}{\partial x_{n}}\big{)}:\big{(}\frac{\partial
f}{\partial x_{1}}\big{)}$ denote the ideal quotient, in the local ring
${\mathcal{O}}_{X,O}$, of $\big{(}f,\frac{\partial f}{\partial x_{2}}$,
$\frac{\partial f}{\partial x_{3}},\ldots,\frac{\partial f}{\partial
x_{n}}\big{)}$ by $\big{(}\frac{\partial f}{\partial x_{1}}\big{)}$. We have
the following.

######  Lemma 3.4.

Let $a(x)$ be a germ of holomorphic function in ${\mathcal{O}}_{X,O}$. Then,
the following are equivalent:

  1. $(i)$

$\displaystyle a(x)\in\bigg{(}f,\frac{\partial f}{\partial
x_{2}},\frac{\partial f}{\partial x_{3}},\ldots,\frac{\partial f}{\partial
x_{n}}\bigg{)}:\bigg{(}\frac{\partial f}{\partial x_{1}}\bigg{)}$.

  2. $(ii)$

There exists a germ of logarithmic vector field $v$ in ${\mathcal{D}}{\rm
er}_{X,O}(-\log S)$ such that

$\displaystyle v=a(x)\frac{\partial}{\partial
x_{1}}+a_{2}(x)\frac{\partial}{\partial
x_{2}}+\cdots+a_{n-1}(x)\frac{\partial}{\partial
x_{n-1}}+a_{n}(x)\frac{\partial}{\partial x_{n}},$

where $a_{2}(x),\ldots,a_{n}(x)\in{\mathcal{O}}_{X,O}$.

Note that in [24, 27], by utilizing local cohomology and Grothendieck local
duality, an effective method of computing a set of generators over the local
ring ${\mathcal{O}}_{X,O}$ of the module of logarithmic vector fields is
given. See the next section.

######  Lemma 3.5.

Assume that $f,\frac{\partial f}{\partial x_{2}},\frac{\partial f}{\partial
x_{3}},\ldots,\frac{\partial f}{\partial x_{n}}$ is a regular sequence. Let
$v^{\prime}$ be a logarithmic vector fields in ${\mathcal{D}}{\rm
er}_{X,O}(-\log S)$ of the form

$\displaystyle v^{\prime}=a_{2}(x)\frac{\partial}{\partial
x_{2}}+a_{3}(x)\frac{\partial}{\partial
x_{3}}+\cdots+a_{n}(x)\frac{\partial}{\partial x_{n}}.$

Then, $v^{\prime}$ is trivial.

Lemmas 3.4 and 3.5 immediately yield the following.

######  Proposition 3.6.

Let $f,\frac{\partial f}{\partial x_{2}},\frac{\partial f}{\partial
x_{3}},\ldots,\frac{\partial f}{\partial x_{n}}$ be a regular sequence. Let
$v$ be a germ of logarithmic vector field along $S$ of the form

$\displaystyle v={a_{1}(x)\frac{\partial}{\partial
x_{1}}+a_{2}(x)\frac{\partial}{\partial
x_{2}}+\cdots+a_{n-1}(x)\frac{\partial}{\partial
x_{n-1}}+a_{n}(x)\frac{\partial}{\partial x_{n}}}.$

Then, the following conditions are equivalent:

  1. $(i)$

$v$ is trivial,

  2. $(ii)$

$a_{1}(x)\in\big{(}f,\frac{\partial f}{\partial x_{2}},\frac{\partial
f}{\partial x_{3}},\ldots,\frac{\partial f}{\partial x_{n}}\big{)}$.

Therefore, we have the following.

######  Theorem 3.7 ([39]).

${\mathcal{D}}{\rm er}_{X,O}(-\log S)/{\sim}$ is isomorphic to

$\displaystyle\bigg{(}\bigg{(}f,\frac{\partial f}{\partial
x_{2}},\frac{\partial f}{\partial x_{3}},\ldots,\frac{\partial f}{\partial
x_{n}}\bigg{)}:\bigg{(}\frac{\partial f}{\partial
x_{1}}\bigg{)}\bigg{)}\Big{/}\bigg{(}f,\frac{\partial f}{\partial
x_{2}},\frac{\partial f}{\partial x_{3}},\ldots,\frac{\partial f}{\partial
x_{n}}\bigg{)}.$

To be more precise, let $A$ be a basis as a vector space of the quotient

$\displaystyle\bigg{(}\bigg{(}f,\frac{\partial f}{\partial
x_{2}},\frac{\partial f}{\partial x_{3}},\ldots,\frac{\partial f}{\partial
x_{n}}\bigg{)}:\bigg{(}\frac{\partial f}{\partial
x_{1}}\bigg{)}\bigg{)}\Big{/}\bigg{(}f,\frac{\partial f}{\partial
x_{2}},\frac{\partial f}{\partial x_{3}},\ldots,\frac{\partial f}{\partial
x_{n}}\bigg{)}.$

Then the corresponding logarithmic vector fields,

$\displaystyle v=a(x)\frac{\partial}{\partial
x_{1}}+a_{2}(x)\frac{\partial}{\partial
x_{2}}+\cdots+a_{n-1}(x)\frac{\partial}{\partial
x_{n-1}}+a_{n}(x)\frac{\partial}{\partial x_{n}},\qquad a(x)\in A$

give rise to a basis of ${\mathcal{D}}{\rm er}_{X,O}(-\log S)/{\sim}$.

###  3.3 Local cohomology and duality

In this section, we briefly recall some basics on local cohomology and
Grothendieck local duality. We give an outline for computing non-trivial
logarithmic vector fields. We refer to [40] for details.

Let ${\mathcal{H}}_{\\{O\\}}^{n}\big{(}\Omega_{X}^{n}\big{)}$ denote the local
cohomology supported at the origin $O$ of the sheaf $\Omega_{X}^{n}$ of
holomorphic $n$-forms. Then, the stalk ${\mathcal{O}}_{X,O}$ and the local
cohomology ${\mathcal{H}}_{\\{O\\}}^{n}\big{(}\Omega_{X}^{n}\big{)}$ are
mutually dual as locally convex topological vector spaces.

The duality is given by the point residue pairing:

${\rm Res}_{\\{O\\}}(*,*)\colon\
{\mathcal{O}}_{X,O}\times{\mathcal{H}}_{\\{O\\}}^{n}\big{(}\Omega_{X}^{n}\big{)}\longrightarrow{\mathbb{C}}.$

Let $W_{\Gamma(f)}$ denote the set of local cohomology classes in
${\mathcal{H}}_{\\{O\\}}^{n}\big{(}\Omega_{X}^{n}\big{)}$ that are annihilated
by $f$, $\frac{\partial f}{\partial x_{2}},\frac{\partial f}{\partial
x_{3}},\dots,\frac{\partial f}{\partial x_{n}}$:

$W_{\Gamma(f)}=\left\\{{\varphi\in{\mathcal{H}}_{\\{O\\}}^{n}\big{(}\Omega_{X}^{n}\big{)}}\left|\,\vphantom{\varphi\in{\mathcal{H}}_{\\{O\\}}^{n}\big{(}\Omega_{X}^{n}\big{)}}{f\varphi=\frac{\partial
f}{\partial x_{2}}\varphi=\cdots=\frac{\partial f}{\partial
x_{n}}\varphi=0}\right.\right\\}.$

Then, a complex analytic version of Grothendieck local duality on residue
implies that the pairing

$\displaystyle{\mathcal{O}}_{X,O}\Big{/}\bigg{(}f,\frac{\partial f}{\partial
x_{2}},\frac{\partial f}{\partial x_{3}},\ldots,\frac{\partial f}{\partial
x_{n}}\bigg{)}\times W_{\Gamma(f)}\longrightarrow{\mathbb{C}}$

is non-degenerate.

Let $\mu(f)$ and $\mu(f|_{H_{x_{1}}})$ denote the Milnor number of $f$ and
that of a hyperplane section $f|_{H_{x_{1}}}$ of $f$, where $f|_{H_{x_{1}}}$
is the restriction of $f$ to the hyperplane $H_{x_{1}}=\\{x\in
X\,|\,x_{1}=0\\}$. Then, the classical Lê–Teissier formula [17, 43] and the
Grothendieck local duality imply the following:

$\dim_{{\mathbb{C}}}W_{\Gamma(f)}=\mu(f)+\mu(f|_{H_{x_{1}}}).$

Let $\gamma\colon W_{\Gamma(f)}\longrightarrow W_{\Gamma(f)}$ be a map defined
by $\gamma(\varphi)=\frac{\partial f}{\partial x_{1}}\ast\varphi$ and let
$W_{\Gamma(f)}$ be the image of the map $\gamma$:

$W_{\Delta(f)}=\left\\{{\frac{\partial f}{\partial
x_{1}}\ast\varphi}\left|\,\vphantom{\frac{\partial f}{\partial
x_{1}}\ast\varphi}{\varphi\in W_{\Gamma(f)}}\right.\right\\}.$

Let ${\rm Ann}_{{\mathcal{O}}_{X,O}}(W_{\Delta(f)})$ be the annihilator in
${\mathcal{O}}_{X,O}$ of the set $W_{\Delta(f)}$ of local cohomology classes.
We have the following.

######  Lemma 3.8 ([39]).

${\rm Ann}_{{\mathcal{O}}_{X,O}}(W_{\Delta(f)})=\big{(}f,\frac{\partial
f}{\partial x_{2}},\frac{\partial f}{\partial x_{3}},\ldots,\frac{\partial
f}{\partial x_{n}}\big{)}:\big{(}\frac{\partial f}{\partial x_{1}}\big{)}$.

###### Proof.

See [20, 39, 41]. ∎

Recall that the ideal quotient $\big{(}f,\frac{\partial f}{\partial
x_{2}},\frac{\partial f}{\partial x_{3}},\ldots,\frac{\partial f}{\partial
x_{n}}\big{)}:\big{(}\frac{\partial f}{\partial x_{1}}\big{)}$ is coefficient
ideal w.r.t. $\frac{\partial}{\partial x_{1}}$ of logarithmic vector fields
along $S$. The lemma above says that the coefficient ideal can be described in
terms of local cohomology $W_{\Delta(f)}$.

Let $W_{T(f)}$ be the kernel of the map $\gamma$. By definition we have

$W_{T(f)}=\left\\{{\varphi\in{\mathcal{H}}_{\\{O\\}}^{n}\big{(}\Omega_{X}^{n}\big{)}}\left|\,\vphantom{\varphi\in{\mathcal{H}}_{\\{O\\}}^{n}\big{(}\Omega_{X}^{n}\big{)}}{f\varphi=\frac{\partial
f}{\partial x_{1}}\varphi=\frac{\partial f}{\partial
x_{2}}\varphi=\cdots=\frac{\partial f}{\partial
x_{n}}\varphi=0}\right.\right\\}.$

Since the pairing

$\displaystyle\displaystyle{\mathcal{O}}_{X,O}\Big{/}\\!\bigg{(}f,\frac{\partial
f}{\partial x_{1}}\frac{\partial f}{\partial x_{2}},\frac{\partial f}{\partial
x_{3}},\ldots,\frac{\partial f}{\partial x_{n}}\bigg{)}\times
W_{T(f)}\longrightarrow{\mathbb{C}}$

is non-degenerate by Grothendieck local duality,
$\dim_{{\mathbb{C}}}(W_{T(f)})$ is equal to

$\tau=\dim_{{\mathbb{C}}}\bigg{(}{\mathcal{O}}_{X,O}\Big{/}\\!\bigg{(}f,\frac{\partial
f}{\partial x_{1}}\frac{\partial f}{\partial x_{2}},\frac{\partial f}{\partial
x_{3}},\ldots,\frac{\partial f}{\partial x_{n}}\bigg{)}\bigg{)},$

the Tjurina number.

From the exactness of the sequence

$0\longrightarrow W_{T(f)}\longrightarrow W_{\Gamma(f)}\longrightarrow
W_{\Delta(f)}\longrightarrow 0,$

we have

$\dim_{{\mathbb{C}}}W_{\Delta(f)}=\mu(f)-\tau(f)+\mu(f|_{H_{x_{1}}}).$

The argument above also implies the following.

######  Corollary 3.9 ([39]).

$\dim_{{\mathbb{C}}}\big{(}{\mathcal{D}}{\rm er}_{X,O}(-\log
S)/{\sim}\big{)}=\tau.$

Notice that the dimension of $W_{\Delta(f)}$ that measures the way of
vanishing of coefficients of logarithmic vector fields depends on the choice
of a system of coordinates, or a hyperplane. In order to analyze complex
analytic properties of logarithmic vector fields, as we observed in [39], it
is important to select an appropriate system of coordinates or a generic
hyperplane. We return to this issue afterwards at the end of this section.

Now let $\displaystyle H_{[O]}^{n}({\mathcal{O}}_{X})=\lim_{k\to\infty}{\rm
Ext}_{{\mathcal{O}}_{X}}^{n}\big{(}{\mathcal{O}}_{X,O}/(x_{1},x_{2},\ldots,x_{n})^{k},{\mathcal{O}}_{X}\big{)}$
be the sheaf of algebraic local cohomology and let

$\displaystyle H_{\Gamma(f)}=\left\\{{\phi\in
H_{[O]}^{n}({\mathcal{O}}_{X})}\left|\,\vphantom{\phi\in
H_{[O]}^{n}({\mathcal{O}}_{X})}{f\phi=\frac{\partial f}{\partial
x_{2}}\phi=\cdots=\frac{\partial f}{\partial x_{n}}\phi=0}\right.\right\\},$
$\displaystyle H_{\Delta(f)}=\left\\{{\frac{\partial f}{\partial
x_{1}}\phi}\left|\,\vphantom{\frac{\partial f}{\partial x_{1}}\phi}{\phi\in
H_{\Gamma(f)}}\right.\right\\}.$

Then, the following holds

$W_{\Gamma(f)}=\\{\phi\cdot\omega_{X}\,|\,\phi\in H_{\Gamma(f)}\\},\qquad
W_{\Delta(f)}=\\{\phi\cdot\omega_{X}\,|\,\phi\in H_{\Delta(f)}\\}.$

In [41], algorithms for computing algebraic local cohomology classes and some
relevant algorithms are given. Accordingly, $H_{\Gamma(f)},H_{\Delta(f)}$ are
computable. Note also that a standard basis of the ideal quotient
$\big{(}f,\frac{\partial f}{\partial x_{2}},\frac{\partial f}{\partial
x_{3}},\ldots,\frac{\partial f}{\partial x_{n}}\big{)}:\big{(}\frac{\partial
f}{\partial x_{1}}\big{)}$ can be computed by using $H_{\Delta(f)}$ in an
efficient manner [41].

Now we present an outline of a method for constructing a basis, as a vector
space, of the quotient space $\big{(}\big{(}f,\frac{\partial f}{\partial
x_{2}},\frac{\partial f}{\partial x_{3}},\ldots,\frac{\partial f}{\partial
x_{n}}\big{)}:\big{(}\frac{\partial f}{\partial
x_{1}}\big{)}\big{)}/\big{(}f,\frac{\partial f}{\partial x_{2}},\frac{\partial
f}{\partial x_{3}},\ldots,\frac{\partial f}{\partial x_{n}}\big{)}$.

We fix a term ordering $\succ$ on $H_{[O]}^{n}({\mathcal{O}}_{X})$ and its
inverse term ordering $\succ^{-1}$ on the local ring ${\mathcal{O}}_{X,O}$.

  1. Step 1:

Compute a basis $\Phi_{\Gamma(f)}$ of $H_{\Gamma(f)}$.

  2. Step 2:

Compute a monomial basis $M_{\Gamma(f)}$ of the quotient space
${\mathcal{O}}_{X,O}/\big{(}f,\frac{\partial f}{\partial x_{2}},\frac{\partial
f}{\partial x_{3}},\ldots,\frac{\partial f}{\partial x_{n}}\big{)}$, with
respect to $\succ^{-1}$, by using $\Phi_{\Gamma(f)}$.

  3. Step 3:

Compute $\frac{\partial f}{\partial x_{n}}\phi$ of each
$\phi\in\Phi_{\Gamma(f)}$ and compute a basis $\Phi_{\Delta(f)}$ of
$H_{\Delta(f)}$.

  4. Step 4:

Compute a standard basis ${\rm SB}$ of the ideal ${\rm
Ann}_{{\mathcal{O}}_{X,O}}(H_{\Delta(f)})$ by using $\Phi_{\Delta(f)}$.

  5. Step 5:

Compute the normal form ${\rm NF}_{\succ^{-1}}\big{(}x^{\lambda}s(x)\big{)}$
of $x^{\lambda}s(x)$ for $x^{\lambda}\in M_{\Gamma(f)},s(x)\in{\rm SB}$.

  6. Step 6:

Compute a basis ${\rm A}$, as a vector space, of ${\rm
Span}_{{\mathbb{C}}}\big{\\{}{\rm
NF}_{\succ^{-1}}(x^{\lambda}s(x))\,|\,x^{\lambda}\in
M_{\Gamma(f)}$,$s(x)\in{\rm SB}\big{\\}}$.

Then, we have the following:

$\displaystyle{\rm Span}_{{\mathbb{C}}}({\rm
A})\cong\bigg{(}\bigg{(}f,\frac{\partial f}{\partial x_{2}},\frac{\partial
f}{\partial x_{3}},\ldots,\frac{\partial f}{\partial
x_{n}}\bigg{)}:\bigg{(}\frac{\partial f}{\partial
x_{1}}\bigg{)}\bigg{)}\Big{/}\bigg{(}f,\frac{\partial f}{\partial
x_{2}},\frac{\partial f}{\partial x_{3}},\ldots,\frac{\partial f}{\partial
x_{n}}\bigg{)}.$

Note that, by utilizing algorithms given in [22], the method proposed above
can be extended to treat parametric cases, the case where the input data
contain parameters.

In order to obtain non-trivial logarithmic vector fields, it is enough to do
the following.

For each $a(x)\in A$, compute
$a_{2}(x),a_{3}(x),\ldots,a_{n}(x),b(x)\in{\mathcal{O}}_{X,O}$, such that

$\displaystyle a(x)\frac{\partial f}{\partial x_{1}}+a_{2}(x)\frac{\partial
f}{\partial x_{2}}+\cdots+a_{n-1}(x)\frac{\partial f}{\partial
x_{n-1}}+a_{n}(x)\frac{\partial f}{\partial x_{n}}-b(x)f(x)=0.$

Then,

$\displaystyle a(x)\frac{\partial}{\partial
x_{1}}+a_{2}(x)\frac{\partial}{\partial
x_{2}}+\cdots+a_{n-1}(x)\frac{\partial}{\partial
x_{n-1}}+a_{n}(x)\frac{\partial}{\partial x_{n}},\qquad a(x)\in A$

gives rise to the desired set of non-trivial logarithmic vector fields.

The step above can be executed efficiently by using an algorithm described in
[21]. See also [40] for details.

Before ending this section, we turn to the issue on the genericity. For this
purpose, let us recall a result of B. Teissier on this subject.

Let $p^{\prime}=(p_{1}^{\prime},p_{2}^{\prime},\dots,p_{n}^{\prime})$ be a
non-zero vector and let $[p^{\prime}]$ denote the corresponding point in the
projective space ${\mathbb{P}}^{n-1}$. We identify the hyperplane

$H_{p^{\prime}}=\big{\\{}(x_{1},x_{2},\dots,x_{n})\in{\mathbb{C}}^{n}\,|\,p_{1}^{\prime}x_{1}+p_{2}^{\prime}x_{2}+\cdots+p_{n}^{\prime}x_{n}=0\big{\\}}$

with the point $[p^{\prime}]$ in ${\mathbb{P}}^{n-1}$. In [43, 44], B.
Teissier introduced an invariant $\mu^{(n-1)}(f)$ as

$\mu^{(n-1)}(f)=\min_{[p^{\prime}]\in{\mathbb{P}}^{n-1}}\mu(f|_{H_{p^{\prime}}}),$

where $f|_{H_{p^{\prime}}}$ is the restriction of $f$ to $H_{p^{\prime}}$ and
$\mu(f|_{H_{p^{\prime}}})$ is the Milnor number at the origin $O$ of the
hyperplane section $f|_{H_{p^{\prime}}}$ of $f$. He also proved that the set

$U=\big{\\{}[p^{\prime}]\in{\mathbb{P}}^{n-1}\,|\,\mu(f|_{H_{p^{\prime}}})=\mu^{(n-1)}(f)\big{\\}}$

is a Zariski open dense subset of ${\mathbb{P}}^{n-1}$.

Accordingly, in order to obtain good representations of logarithmic vector
fields, it is desirable to use a generic system of coordinate or a generic
hyperplane $H_{p^{\prime}}$ that satisfies the condition
$\mu(f|_{H_{p^{\prime}}})=\mu^{(n-1)}(f)$.

In a previous paper [25], methods for computing limiting tangent spaces were
studied and an algorithm of computing $\mu(f|_{H_{p^{\prime}}})$,
$p^{\prime}\in{\mathbb{P}}^{n-1}$ was given. In [23, 26], more effective
algorithms for computing $\mu^{(n-1)}$ were given. Utilizing the results in
[23, 26], an effective method for computing logarithmic vector fields that
takes care of the genericity condition is designed in [27, 40]. See also [42]
for related results.

###  3.4 Regular meromorphic differential forms

Now we are ready to consider a method for computing regular meromorphic
differential forms. For simplicity, we first consider a 3-dimensional case.
Assume that a non-trivial logarithmic vector field $v$ is given:

$\displaystyle v=a_{1}(x)\frac{\partial}{\partial
x_{1}}+a_{2}(x)\frac{\partial}{\partial
x_{2}}+a_{3}(x)\frac{\partial}{\partial x_{3}}.$

Let $v(f)=b(x)f(x)$ and $\beta=i_{v}(\omega_{X})$, where $\omega_{X}={\rm
d}x_{1}\wedge{\rm d}x_{2}\wedge{\rm d}x_{3}$. We have $\beta=a_{1}(x){\rm
d}x_{2}\wedge{\rm d}x_{3}-a_{2}(x){\rm d}x_{1}\wedge{\rm d}x_{3}+a_{3}(x){\rm
d}x_{1}\wedge{\rm d}x_{2}$. We introduce differential forms $\xi$ and $\eta$
as

$\displaystyle\xi=-a_{2}(x){\rm d}x_{3}+a_{3}(x){\rm
d}x_{2},\qquad\eta=b(x){\rm d}x_{2}\wedge{\rm d}x_{3}.$

Let $g(x)=\frac{\partial f}{\partial x_{1}}$. Then, the following holds

$\displaystyle g(x)\beta={\rm d}f\wedge\xi+f(x)\eta.$

Accordingly, the logarithmic differential form $\omega=\frac{\beta}{f}$
satisfies

$\displaystyle g(x)\omega=\frac{{\rm d}f}{f}\wedge\xi+\eta.$

We may assume that the coordinate system $(x_{1},x_{2},x_{3})$ is generic [27]
and $g(x)$ satisfies the condition $(a)$, $(b)$ of $(iii)$ in Definition 2.1.

Since $g(x)=\frac{\partial f}{\partial x_{1}},$ we have, by definition, the
following:

$\displaystyle{\rm
res}\bigg{(}\frac{\beta}{f}\bigg{)}=\frac{\xi}{\frac{\partial f}{\partial
x_{1}}}\Bigg{|}_{S}.$

Notice that the differential form $\xi$ above is directly defined from the
coefficients of the logarithmic vector field $v$.

######  Proposition 3.10.

Let $S=\\{x\in X\,|\,f(x)=0\\}$ be a hypersurface with an isolated singularity
at the origin $O\in X\subset{\mathbb{C}}^{n}$. Assume that the coordinate
system $(x_{1},x_{2},\ldots,x_{n})$ is generic so that
$\big{(}f,\frac{\partial f}{\partial x_{2}},\frac{\partial f}{\partial
x_{3}},\ldots,\frac{\partial f}{\partial x_{n}}\big{)}$ is a regular sequence
and $g(x)=\frac{\partial f}{\partial x_{1}}$ satisfies the condition $(a)$,
$(b)$ of $(iii)$ in Definition 2.1. Let

$\displaystyle v=a_{1}(x)\frac{\partial}{\partial
x_{1}}+a_{2}(x)\frac{\partial}{\partial
x_{2}}+\cdots+a_{n}(x)\frac{\partial}{\partial x_{n}}$

be a germ of non-trivial logarithmic vector field along $S$. Let
$v(f)=b(x)f(x)$, $\beta=i_{v}(\omega_{X})$. Let $\xi$, $\eta$ denote the
differential form defined to be

$\displaystyle\xi=-a_{2}(x){\rm d}x_{3}\wedge{\rm
d}x_{4}\wedge\cdots\wedge{\rm d}x_{n}+a_{3}(x){\rm d}x_{2}\wedge{\rm
d}x_{4}\wedge\cdots\wedge{\rm d}x_{n}-\cdots$
$\displaystyle\phantom{\xi=-}{}+(-1)^{(n+1)}a_{n}(x){\rm d}x_{2}\wedge{\rm
d}x_{3}\wedge\cdots\wedge{\rm d}x_{n-1},$
$\displaystyle\eta=\phantom{-}b(x){\rm d}x_{2}\wedge{\rm
d}x_{3}\wedge\cdots\wedge{\rm d}x_{n}.$

Then,

$\displaystyle g(x)\frac{\beta}{f}=\frac{{\rm
d}f}{f}\wedge\xi+\eta\qquad\text{and}\qquad{\rm
res}\bigg{(}\frac{\beta}{f}\bigg{)}=\frac{\xi}{\frac{\partial f}{\partial
x_{1}}}\Bigg{|}_{S}$

hold.

Note that, in 1984, M. Kersken [16] obtained related results on regular
meromorphic differential forms. The statement in Proposition 3.10 above is a
refinement a result of M. Kersken.

######  Theorem 3.11.

Let $S=\\{x\in X\,|\,f(x)=0\\}$ be a hypersurface with an isolated singularity
at the origin $O\in X\subset{\mathbb{C}}^{n}$. Let
$V=\\{v_{1},v_{2},\ldots,v_{\tau}\\}$ be a set of non-trivial logarithmic
vector fields such that the class $[v_{1}],[v_{2}],\dots,[v_{\tau}]$
constitute a basis of the vector space ${\mathcal{D}}{\rm er}_{X,O}(-\log
S)/{\sim}$, where $\tau$ stands for the Tjurina number of $f$. Let
$\xi_{1},\xi_{2},\ldots,\xi_{\tau}$ be the differential forms correspond to
$v_{1},v_{2},\ldots,v_{\tau}$ defined in Proposition 3.10.

Then, any logarithmic residue in ${\rm res}\big{(}\Omega^{n-1}(\log
S)\big{)}$, or a regular meromorphic differential form $\gamma$ in
$\omega_{S}^{n-2}$ can be represented as

$\displaystyle\gamma=\bigg{(}\frac{1}{\frac{\partial f}{\partial
x_{1}}}(c_{1}\xi_{1}+c_{2}\xi_{2}+\cdots+c_{\tau}\xi_{\tau})\bigg{)}\bigg{|}_{S}+\alpha,$

where $c_{i}\in{\mathbb{C}}$, $i=1,2,\dots,\tau$, and
$\left.\alpha\in\Omega_{X}^{n-2}\middle|\raisebox{-3.01385pt}[4.30554pt][-8.61108pt]{{\;${}_{S}$}}\right.$.

##  4 Examples

In this section, we give examples of computation for illustration. Data is an
extraction from [40]. Let $f_{0}(z,x,y)=x^{3}+y^{3}+z^{4}$ and let
$f_{t}(z,x,y)=f_{0}(z,x,y)+txyz^{2}$, where $t$ is a deformation parameter. We
regard $z$ as the first variable. Then, $f_{0}$ is a weighted homogeneous
polynomial with respect to a weight vector $(3,4,4)$ and $f_{t}$ is a
$\mu$-constant deformation of $f_{0}$, called $U_{12}$ singularity. The Milnor
number $\mu(f_{t})$ of $U_{12}$ singularity is equal to 12. In contrast, the
Tjurina number $\tau(f_{t})$ depends on the parameter $t$. In fact, if $t=0$,
then $\tau(f_{0})=12$ and if $t\neq 0$, then $\tau(f_{t})=11$. In the
computation, we fix a term order $\succ^{-1}$ on ${\mathcal{O}}_{X,O}$ which
is compatible with the weight vector $(3,4,4)$.

We consider these two cases separately.

######  Example 4.1 (weighted homogeneous $U_{12}$ singularity).

Let $f_{0}(z,x,y)=x^{3}+y^{3}+z^{4}$. Then, $\mu(f_{0})=\tau(f_{0})=12$. The
monomial basis ${\rm M}$ with respect to the term ordering $\succ^{-1}$ of the
quotient space ${\mathcal{O}}_{X,O}/(f_{0},\frac{\partial f_{0}}{\partial
x},\frac{\partial f_{0}}{\partial y})$ is

$\displaystyle{\rm M}=\big{\\{}x^{i}y^{j}z^{k}\,|\,i=0,1,\ j=0,1,\
k=0,1,2,3\big{\\}}.$

The standard basis ${\rm Sb}$ of the ideal quotient
$\big{(}f_{0},\frac{\partial f_{0}}{\partial x},\frac{\partial f_{0}}{\partial
y}\big{)}:\big{(}\frac{\partial f_{0}}{\partial z}\big{)}$ is

$\displaystyle{\rm Sb}=\big{\\{}x^{2},y^{2},z\big{\\}}.$

The normal form in ${\mathcal{O}}_{X,O}/\big{(}f_{0},\frac{\partial
f_{0}}{\partial x},\frac{\partial f_{0}}{\partial y}\big{)}$ of $x^{2}$,
$y^{2}$ and $z$ are

$\displaystyle{\rm NF}_{\succ^{-1}}\big{(}x^{2}\big{)}={\rm
NF}_{\succ^{-1}}\big{(}y^{2}\big{)}=0,\qquad{\rm NF}_{\succ^{-1}}(z)=z.$

Therefore, ${\rm A}=\\{x^{i}y^{j}z^{k}\,|\,i=0,1,\ j=0,1,\ k=1,2,3\\}$. Notice
that ${\rm A}$ consists of $12$ elements. It is easy to see that the Euler
vector field

$\displaystyle v=4x\frac{\partial}{\partial x}+4y\frac{\partial}{\partial
y}+3z\frac{\partial}{\partial z}$

that corresponds to the element $z\in{\rm A}$ is a non-trivial logarithmic
vector field. Therefore, the torsion module of the hypersurface
$S_{0}=\big{\\{}(x,y,z)\,|\,x^{3}+y^{3}+z^{4}=0\big{\\}}$ is given by

$\displaystyle{\rm
Tor}\big{(}\Omega_{S_{0}}^{2}\big{)}=\big{\\{}x^{i}y^{j}z^{k}i_{v}(\omega_{X})\,|\,i=0,1,\
j=0,1,\ k=1,2,3\big{\\}},$

where $\omega_{X}={\rm d}z\wedge{\rm d}x\wedge{\rm d}y$.

Let $\xi=-4x{\rm d}y+4y{\rm d}x$. Then $\left.{\rm
res}\big{(}\frac{i_{v}(\omega_{X})}{f}\big{)}=\frac{\xi}{4z^{3}}\middle|\raisebox{-3.01385pt}[4.30554pt][-8.61108pt]{{\;${}_{S}$}}\right.$.
Computation of other logarithmic residues are same.

The following is also an extraction from [40].

######  Example 4.2 (semi quasi-homogeneous $U_{12}$ singularity).

Let $f(x,y,z)=x^{3}+y^{3}+z^{4}+txyz^{2}$, $t\neq 0$. Then, $\mu(f)=12$,
$\tau(f)=11$ and $\mu(f|H_{z})=4$. We have
$\dim_{{\mathbb{C}}}H_{\Gamma(f)}=16$, $\dim_{{\mathbb{C}}}H_{\Delta(f)}=5$.
Let $\succ$ be a term ordering on $H_{[O]}^{3}({\mathcal{O}}_{X})$ which is
compatible with the weight vector $(4,4,3)$.

A basis $\Phi_{{\Gamma(f)}}$ of $H_{{\Gamma(f)}}$ is given by

$\displaystyle\left\\{\begin{bmatrix}1\\\ xyz\end{bmatrix},\begin{bmatrix}1\\\
xyz^{2}\end{bmatrix},\begin{bmatrix}1\\\
x^{2}yz\end{bmatrix},\begin{bmatrix}1\\\
xy^{2}z\end{bmatrix},\begin{bmatrix}1\\\
xyz^{3}\end{bmatrix},\begin{bmatrix}1\\\
x^{2}yz^{2}\end{bmatrix},\begin{bmatrix}1\\\
xy^{2}z^{2}\end{bmatrix},\begin{bmatrix}1\\\
x^{2}y^{2}z\end{bmatrix},\begin{bmatrix}1\\\ xyz^{4}\end{bmatrix},\right.$
$\displaystyle\hphantom{\left\\{\right.}\begin{bmatrix}1\\\
x^{2}yz^{3}\end{bmatrix}-\dfrac{t}{3}\begin{bmatrix}1\\\
xy^{3}z\end{bmatrix},\begin{bmatrix}1\\\
xy^{2}z^{3}\end{bmatrix}-\dfrac{t}{3}\begin{bmatrix}1\\\
x~{}3yz\end{bmatrix},\begin{bmatrix}1\\\
x^{2}y^{2}z^{2}\end{bmatrix},\begin{bmatrix}1\\\
x^{2}yz^{4}\end{bmatrix}-\dfrac{t}{3}\begin{bmatrix}1\\\
xy^{3}z^{2}\end{bmatrix},$
$\displaystyle\hphantom{\left\\{\right.}\begin{bmatrix}1\\\
xy^{2}z^{4}\end{bmatrix}-\dfrac{t}{3}\begin{bmatrix}1\\\
x^{3}yz^{2}\end{bmatrix},\begin{bmatrix}1\\\
x^{2}y^{2}z^{3}\end{bmatrix}-\dfrac{t}{3}\begin{bmatrix}1\\\
x^{4}yz\end{bmatrix}-\dfrac{t}{3}\begin{bmatrix}1\\\
xy^{4}z\end{bmatrix}-\dfrac{t}{3}\begin{bmatrix}1\\\ xyz^{5}\end{bmatrix},$
$\displaystyle\hphantom{\left\\{\right.}\left.\begin{bmatrix}1\\\
x^{2}y^{2}z^{4}\end{bmatrix}-\dfrac{t}{3}\begin{bmatrix}1\\\
x^{4}yz^{2}\end{bmatrix}-\dfrac{t}{3}\begin{bmatrix}1\\\
xy^{4}z^{2}\end{bmatrix}-\dfrac{t}{3}\begin{bmatrix}1\\\
xyz^{6}\end{bmatrix}\right\\}.$

The monomial basis ${\rm M}$ with respect to the term ordering $\succ^{-1}$ of
the quotient ${\mathcal{O}}_{X,O}/\big{(}f,$ $\frac{\partial f}{\partial
x},\frac{\partial f}{\partial y}\big{)}$ is

$\displaystyle{\rm M}=\big{\\{}x^{i}y^{j}z^{k}\,|\,i=0,1,\ j=0,1,\
k=0,1,2,3\big{\\}}.$

A basis $\Phi_{\Delta(f)}$ of $H_{\Delta(f)}$ is given by

$\left\\{\begin{bmatrix}1\\\ xyz\end{bmatrix},\begin{bmatrix}1\\\
xyz^{2}\end{bmatrix},\begin{bmatrix}1\\\
x^{2}yz\end{bmatrix},\begin{bmatrix}1\\\
xy^{2}z\end{bmatrix},\begin{bmatrix}1\\\
x^{2}y^{2}z\end{bmatrix}+\dfrac{t}{6}\begin{bmatrix}1\\\
xyz^{3}\end{bmatrix}\right\\}.$

We see from this data that the standard basis of the ideal quotient
$\big{(}f,\frac{\partial f}{\partial x},\frac{\partial f}{\partial
y}\big{)}:\big{(}\frac{\partial f}{\partial z}\big{)}$ in the local ring
${\mathcal{O}}_{X,O}$ is

$\displaystyle{\rm
Sb}=\left\\{z^{2}-\dfrac{t}{6}xy,xz,yz,x^{2},y^{2}\right\\}.$

From ${\rm Sb}$ and ${\rm M}$, we have

$\displaystyle{\rm
A}=\left\\{z^{2}-\dfrac{t}{6}xy,xz,yz,z^{3},xz^{2},yz^{2},xyz,xz^{3},yz^{3},xyz^{2},xyz^{3}\right\\}.$

These 11 elements in ${\rm A}$ are used to construct non-trivial logarithmic
vector fields and regular meromorphic differential forms. We give the results
of computation.

  1. $(i)$

Let $a=6z^{2}-txy$. Then,

$\displaystyle v=\frac{d_{1}}{27+t^{3}z^{2}}\frac{\partial}{\partial
x}+\frac{d_{2}}{27+t^{3}z^{2}}\frac{\partial}{\partial
y}+\big{(}6z^{2}-txy\big{)}\frac{\partial}{\partial z}$

is a non-trivial logarithmic vector field, where

$\displaystyle d_{1}=216xz-6t^{2}y^{2}z-2t^{4}x^{2}yz,\qquad
d_{2}=216yz+24t^{2}x^{2}z+10t^{3}yz^{3}-2t^{4}xy^{2}z.$

  2. $(ii)$

Let $a=xz$. Then,

$\displaystyle v=\frac{d_{1}}{27+t^{3}z^{2}}\frac{\partial}{\partial
x}+\frac{d_{2}}{27+t^{3}z^{2}}\frac{\partial}{\partial
y}+xz\frac{\partial}{\partial z}$

is a non-trivial logarithmic vector field, where

$\displaystyle d_{1}=36x^{2}-6yz^{2}-6t^{2}xy^{2},\qquad
d_{2}=36xy+2t^{2}x^{3}-4t^{2}y^{3}-2t^{2}z^{4}.$

We omit the other nine cases. As described in Theorem 3.11, regular
meromorphic differential forms can be constructed directly from these data.

##  5 Brieskorn formula

In 1970, B. Brieskorn studied the monodromy of Milnor fibration and developed
the theory of Gauss–Manin connection [7]. He proved the regularity of the
connection and proposed an algebraic framework for computing the monodromy via
Gauss–Manin connection. He gave in particular a basic formula, now called
Brieskorn formula, for computing Gauss–Manin connection.

We show in this section a link between Brieskorn formula, torsion differential
forms and logarithmic vector fields. We present an alternative method for
computing non-trivial logarithmic vector fields. The resulting algorithm can
be used as a basic tool for studying Gauss–Manin connections. We also present
some examples for illustration.

###  5.1 Brieskorn lattice and Gauss–Manin connection

We briefly recall some basics on Brieskorn lattice and Brieskorn formula. We
refer to [6, 7, 37]. Let $f(x)$ be a holomorphic function on $X$ with an
isolated singularity at the origin $O\in X,$ where $X$ is an open neighborhood
of $O$ in ${\mathbb{C}}^{n}$. Let

$\displaystyle H_{0}^{\prime}=\Omega_{X,O}^{n-1}/\big{(}{\rm
d}f\wedge\Omega_{X,O}^{n-2}+{\rm d}\Omega_{X,O}^{n-2}\big{)},\qquad
H_{0}^{\prime\prime}=\Omega_{X,O}^{n}/{\rm d}f\wedge{\rm
d}\Omega_{X,O}^{n-2}.$

Then, ${\rm d}f\wedge H_{0}^{\prime}\subset H_{0}^{\prime\prime}$. A map
$D\colon{\rm d}f\wedge H_{0}^{\prime}\longrightarrow H_{0}^{\prime\prime}$ is
defined as follows:

$\displaystyle D({\rm d}f\wedge\varphi)=[{\rm
d}\varphi],\qquad\varphi\in\Omega_{X,O}^{n-1}.$

Let $\varphi=\sum_{i=1}^{n}(-1)^{i+1}h_{i}(x){\rm d}x_{1}\wedge{\rm
d}x_{2}\wedge\cdots\wedge{\rm d}x_{i-1}\wedge{\rm
d}x_{i+1}\wedge\cdots\wedge{\rm d}x_{n}$. Then

$\displaystyle{\rm
d}f\wedge\varphi=\Bigg{(}\sum_{i=1}^{n}h_{i}(x)\frac{\partial f}{\partial
x_{i}}\Bigg{)}\omega_{X},$

where $\omega_{X}={\rm d}x_{1}\wedge{\rm d}x_{2}\wedge\cdots\wedge{\rm
d}x_{n}$. Therefore in terms of the coordinate we have the following, known as
Brieskorn formula

$\displaystyle D({\rm d}f\wedge\varphi)=\Bigg{(}\sum_{i}^{n}\frac{\partial
h_{i}}{\partial x_{i}}\Bigg{)}\omega_{X}.$

######  Example 5.1.

Let $f(x,y)=x^{2}-y^{3}$ and $S=\\{(x,y)\in X\,|\,f(x,y)=0\\}$ where
$X\subset{\mathbb{C}}^{2}$ is an open neighborhood of the origin $O$. The
Jacobi ideal $J$ of $f$ is $\big{(}x,y^{2}\big{)}\subset{\mathcal{O}}_{X,O}$
and $M=\\{1,y\\}$ is a monomial basis of the quotient ${\mathcal{O}}_{X,O}/J$.
Let $\tau$ denote the Tjurina number. Then, since $f$ is a weighted
homogeneous polynomial, we have $\tau=\mu=2$ (see Example 2.5).

Let $v=\frac{1}{6}\big{(}3x\frac{\partial}{\partial
x}+2y\frac{\partial}{\partial y}\big{)}$ be the Euler vector field. Then, $v$
is logarithmic along $S$.

Let $\beta=i_{v}(\omega_{X})$. Then, $\beta=\frac{1}{6}(3x{\rm d}y-2y{\rm
d}x)$. Since $v(f)=f$, we have ${\rm d}f\wedge\beta=f\omega_{X},$ where
$\omega_{X}={\rm d}x\wedge{\rm d}y$. By Brieskorn formula, we have

$\displaystyle D(f\omega_{X})=D({\rm d}f\wedge\beta)=\dfrac{5}{6}\omega_{X}.$

Note that the formula above is equivalent ${\rm
d}\big{(}\frac{\beta}{f^{\lambda}}\big{)}=0$, with $\lambda=\frac{5}{6}$.

Likewise, for $y\beta$, we have ${\rm d}f\wedge(y\beta)=f(x,y)y\omega_{X}$ and

$\displaystyle D(f(x,y)y\omega_{X})=D({\rm
d}f\wedge(y\beta))=\dfrac{7}{6}y\omega_{X},$

which is equivalent to ${\rm d}\big{(}\frac{y\beta}{f^{\lambda}}\big{)}=0$,
with $\lambda=\frac{7}{6}$.

Since $Df=fD+1$ as operators, we have

$fD(\omega_{X})=-\dfrac{1}{6}\omega_{X},\qquad
fD(y\omega_{X})=\dfrac{1}{6}y\omega_{X}.$

Notice that $\beta$, $y\beta$ are non-zero torsion differential forms in
$\Omega_{S}^{1}$ and $v$, $yv$ are non-trivial logarithmic vector fields along
$S$. Note also that $yv(f)=yf$. Notably, Brieskorn formula described in terms
of differential forms can be rewritten in terms of non-trivial logarithmic
vector fields $v$ and $yv$ which satisfy $v(f)=f$ and $yv(f)=yf$ respectively.

Let $S=\\{x\in X\,|\,f(x)=0\\}$ be the hypersurface with an isolated
singularity at the origin $O\in X$ defined by $f$. Consider, for instance, a
trivial vector field $v^{\prime}=\frac{\partial f}{\partial
x_{2}}\frac{\partial}{\partial x_{1}}-\frac{\partial f}{\partial
x_{1}}\frac{\partial}{\partial x_{2}}$. Since $v^{\prime}(f)=0$ and
$\frac{\partial}{\partial x_{1}}\big{(}\frac{\partial f}{\partial
x_{2}}\big{)}+\frac{\partial}{\partial x_{2}}\big{(}{-}\frac{\partial
f}{\partial x_{1}}\big{)}=0$ hold, we have a trivial relation
$D((0\cdot\omega_{X})=0\cdot\omega_{X}$. It is easy to see in general that,
from a trivial vector field Brieskorn formula only gives the trivial relation.

The observation above leads the following.

######  Proposition 5.2.

Let $S=\\{x\in X\,|\,f(x)=0\\}$ be a hypersurface with an isolated singularity
at the origin $O\in X,$ where $X\subset{\mathbb{C}}^{n}$. Let

$\displaystyle v=a_{1}(x)\frac{\partial}{\partial
x_{1}}+a_{2}(x)\frac{\partial}{\partial
x_{2}}+\cdots+a_{n}(x)\frac{\partial}{\partial x_{n}}$

be a germ of non-trivial logarithmic vector field along $S$. Let
$v(f)=b(x)f(x)$ Then,

$\displaystyle D(f(x)b(x)\omega_{X})=\Bigg{(}\sum_{i=1}^{n}\frac{\partial
a_{i}}{\partial x_{i}}\Bigg{)}\omega_{X}$

holds, where $\omega_{X}={\rm d}x_{1}\wedge{\rm d}x_{2}\wedge\cdots\wedge{\rm
d}x_{n}$.

###### Proof.

Let $\beta=i_{v}(\omega_{X})$. Since ${\rm d}f\wedge\beta=v(f)\omega_{X},$ we
have ${\rm d}f\wedge\beta=\left(\sum_{i=1}^{n}a_{i}(x)\frac{\partial
f}{\partial x_{i}}\right)\omega_{X}$. Since $v(f)=b(x)f(x),$ Brieskorn formula
implies the result. ∎

Notice that the action of $Df$ on $b(x)\omega_{X}$ in the formula above is
completely written in terms of non-trivial logarithmic vector field $v$ such
that $v(f)=b(x)f$. To the best of our knowledge, this simple observation has
not been explicitly stated in literature on Gauss–Manin connections.

Now we present an alternative method for computing the module of germs of non-
trivial logarithmic vector fields.

  1. Step 1:

Compute a monomial basis ${\rm M}$ of the quotient space

$\displaystyle{\mathcal{O}}_{X,O}\Big{/}\bigg{(}\frac{\partial f}{\partial
x_{1}},\frac{\partial f}{\partial x_{2}},\dots,\frac{\partial f}{\partial
x_{n}}\bigg{)}.$

  2. Step 2:

Compute a standard basis ${\rm Sb}$ of the ideal quotient

$\displaystyle\bigg{(}\frac{\partial f}{\partial x_{1}},\frac{\partial
f}{\partial x_{2}},\dots,\frac{\partial f}{\partial x_{n}}\bigg{)}:(f).$

  3. Step 3:

Compute a basis ${\rm B}$ of the vector space by using ${\rm Sb}$ and ${\rm
M}$

$\displaystyle\bigg{(}\bigg{(}\frac{\partial f}{\partial x_{1}},\frac{\partial
f}{\partial x_{2}},\dots,\frac{\partial f}{\partial
x_{n}}\bigg{)}:(f)\bigg{)}\Big{/}\bigg{(}\frac{\partial f}{\partial
x_{1}},\frac{\partial f}{\partial x_{2}},\dots,\frac{\partial f}{\partial
x_{n}}\bigg{)}.$

  4. Step 4:

For each $b(x)\in{\rm B},$ compute a logarithmic vector field along $S$ such
that

$\displaystyle v(f)=b(x)f(x).$

The method above computes a basis of non-trivial logarithmic vector fields.
Each step can be effectively executable, as in [40], by utilizing algorithms
described in [20, 21, 22, 41].

Note that, the number of non-trivial logarithmic vector fields in the output
is equals to the Tjurina number $\tau(f)$. See also [18].

Let

$\displaystyle v=a_{1}(x)\frac{\partial}{\partial
x_{1}}+a_{2}(x)\frac{\partial}{\partial
x_{2}}+\cdots+a_{n}(x)\frac{\partial}{\partial x_{n}}$

be a germ of non-trivial logarithmic vector field along $S,$ such that
$v(f)=b(x)f(x)$. Then from Proposition 5.2, we have

$\displaystyle D(f(x)b(x)\omega_{X})=\left(\sum_{i=1}^{n}\frac{\partial
a_{i}}{\partial x_{i}}\right)\omega_{X}.$

Therefore, the proposed method can be used as a basic procedure for computing
a connection matrix of Gauss–Manin connection.

One of the advantages of the proposed method lies in the fact that the
resulting algorithm also can handle parametric cases.

###  5.2 Examples

Let us recall that $x^{3}+y^{7}+txy^{5}$ is the standard normal form of semi
quasi-homogeneous $E_{12}$ singularity. The weight vector is $(7,3)$ and the
weighted degree of the quasi-homogeneous part is equal to $21$ and the
weighted degree of the upper monomial $txy^{5}$ is equal to $22$. We examine
here, by contrast, the case where the weighted degree of an upper monomial is
bigger than $22.$

######  Example 5.3.

Let $f(x,y)=x^{3}+y^{7}+txy^{6},$ where $t$ is a parameter. Notice that the
polynomial $f$ is not weighted homogeneous. The weighted degree of the upper
monomial $txy^{6}$ is equal to $25$, which is bigger than that of $txy^{5}$.
Accordingly $f$ is a quasi homogeneous function. The Milnor number $\mu$ is
equal to $12$.

Let $H_{J}$ denote the set of local cohomology classes in
$H_{[0,0]}^{2}({\mathcal{O}}_{X})$ that are killed by the Jacobi ideal
$J=\big{(}\frac{\partial f}{\partial x},\frac{\partial f}{\partial y}\big{)}$:

$\displaystyle H_{J}=\left\\{\psi\in
H_{[0,0]}^{2}({\mathcal{O}}_{X})\,\middle|\,\frac{\partial f}{\partial
x}\psi=\frac{\partial f}{\partial y}\psi=0\right\\}.$

Then, by using an algorithm given in [22, 41], a basis as a vector space of
$H_{J}$ is computed as

$\displaystyle\begin{bmatrix}1\\\ xy\end{bmatrix},\begin{bmatrix}1\\\
xy^{2}\end{bmatrix},\begin{bmatrix}1\\\
xy^{3}\end{bmatrix},\begin{bmatrix}1\\\
x^{2}y\end{bmatrix},\begin{bmatrix}1\\\
xy^{4}\end{bmatrix},\begin{bmatrix}1\\\
x^{2}y^{2}\end{bmatrix},\begin{bmatrix}1\\\
xy^{5}\end{bmatrix},\begin{bmatrix}1\\\
x^{2}y^{3}\end{bmatrix},\begin{bmatrix}1\\\
xy^{6}\end{bmatrix},\begin{bmatrix}1\\\
x^{2}y^{4}\end{bmatrix},\begin{bmatrix}1\\\ x^{2}y^{5}\end{bmatrix},$
$\displaystyle\begin{bmatrix}1\\\
x^{2}y^{6}\end{bmatrix}-\dfrac{6}{7}t\begin{bmatrix}1\\\
xy^{7}\end{bmatrix}+\dfrac{2}{7}t^{2}\begin{bmatrix}1\\\ x^{3}y\end{bmatrix},$

where $[\ ]$ stands for Grothendieck symbol.

It is easy to see that every local cohomology classes in $H_{J}$ is killed by
$f$, that is $f\cdot\varphi=0$, $\forall\,\varphi\in H_{J}$. Therefore, $f$ is
in the ideal $\big{(}\frac{\partial f}{\partial x},\frac{\partial f}{\partial
y}\big{)}\subset{\mathcal{O}}_{X,O}$.

Therefore, by a classical result of K. Saito [31], $f$ is in fact quasi-
homogeneous. The Tjurina number $\tau$ is equal to the Milnor number $\mu=12$.
A monomial basis ${\rm M}$ of ${\mathcal{O}}_{X,O}/\big{(}\frac{\partial
f}{\partial x},\frac{\partial f}{\partial y}\big{)}$ is

$\displaystyle{\rm
M}=\big{\\{}1,y,y^{2},x,y^{3},xy,y^{4},xy^{2},y^{5},xy^{3},xy^{4},xy^{5}\big{\\}}.$

Since a standard basis ${\rm Sb}$ of $\big{(}\frac{\partial f}{\partial
x},\frac{\partial f}{\partial y}\big{)}:(f)$ is $\\{1\\},$ a basis ${\rm B}$
of the vector space $\big{(}\big{(}\frac{\partial f}{\partial
x},\frac{\partial f}{\partial y}\big{)}:(f)\big{)}/\big{(}\frac{\partial
f}{\partial x},\frac{\partial f}{\partial y}\big{)}$ is equal to ${\rm M}$
that consists of $\tau=12$ elements.

By using an algorithm given in [21], we compute a logarithmic vector field
which plays the role of Euler vector field. The result of computation is the
following:

$\displaystyle
v=\frac{d_{1}}{3\big{(}49+12t^{3}y^{4}\big{)}}\frac{\partial}{\partial
x}+\frac{d_{2}}{3\big{(}49+12t^{3}y^{4}\big{)}}\frac{\partial}{\partial y},$

where

$\displaystyle d_{1}=49x+8t^{2}y^{5}+12t^{3}xy^{4},\qquad
d_{2}=21y-4tx+4t^{3}y^{5}.$

The vector field $v$ enjoys $v(f)=f$. Note also that for the case $t=0,$ we
have

$\displaystyle v=\frac{1}{21}\bigg{(}7x\frac{\partial}{\partial
x}+3y\frac{\partial}{\partial y}\bigg{)}.$

We emphasize here the fact that, the algorithm in [27] for computing
logarithmic vector fields can handle parametric cases. Since $v(f)=f$ holds,
the other non-trivial logarithmic vector fields can be obtained from $v$. In
fact, for $x^{i}y^{j}\in M$, we have $x^{i}y^{j}v(f)=x^{i}y^{j}f$.

Therefore, thanks to Brieskorn formula, Gauss–Manin connection can be
determined explicitly by using these non-trivial logarithmic vector fields,

######  Remark 5.4.

Recall that, according to Grothendieck local duality theorem, the vector space
$H_{J}$ can be regarded as a dual space to ${\mathcal{O}}_{X,O}/J$. Since
these local cohomology classes given above constitute a dual basis of the
monomial basis M of the quotient space ${\mathcal{O}}_{X,O}/J$, the normal
form of a holomorphic function w.r.t. ${\mathcal{O}}_{X,O}/J$ can be computed
by using the basis of $H_{J}$ in an efficient manner, without using division
algorithms [41].

Therefore the use of local cohomology classes in reduction steps allows us to
design an effective procedure for computing the connection matrix of
Gauss–Manin connection.

J. Scherk studied in [35] the following case.

######  Example 5.5.

Let $f(x,y)=x^{5}+x^{2}y^{2}+y^{5}$. Then, the Milnor number $\mu(f)$ is equal
to 11 and the Tjurina number $\tau(f)$ is equal to 10. A monomial basis ${\rm
M}$ of ${\mathcal{O}}_{X,O}/\big{(}\frac{\partial f}{\partial
x},\frac{\partial f}{\partial y}\big{)}$ is ${\rm
M}=\big{\\{}1,x,x^{2},x^{3},x^{4},x^{5},xy,y,y^{2},y^{3},y^{4}\big{\\}}$. A
standard basis ${\rm Sb}$ of the ideal quotient $\big{(}\frac{\partial
f}{\partial x}\frac{\partial f}{\partial y}\big{)}:(f)$ is $\\{x,y\\}$. A
basis ${\rm B}$ of the vector space $\big{(}\big{(}\frac{\partial f}{\partial
x},\frac{\partial f}{\partial y}\big{)}:(f)\big{)}/\big{(}\frac{\partial
f}{\partial x},\frac{\partial f}{\partial y}\big{)}$ is

$\displaystyle{\rm
B}=\big{\\{}x,x^{2},x^{3},x^{4},x^{5},xy,y,y^{2},y^{3},y^{4}\big{\\}}.$

Since ${\rm Sb}\cap{\rm B}=\\{x,y\\}$, we first compute non-trivial
logarithmic vector fields associated to $x$ and $y$.

  1. $(i)$

For $b(x,y)=x$, we have

$\displaystyle v=\frac{d_{1}}{5(4-25xy)}\frac{\partial}{\partial
x}+\frac{d_{2}}{5(4-25xy)}\frac{\partial}{\partial y},$

where $d_{1}=4x^{2}-25x^{3}y-5y^{3}$, $d_{2}=6xy-25x^{2}y^{2}$.

Since $v(f)=xf$, by a direct computation, we have for instance

$\displaystyle D(f(x,y)x\omega_{X})=\left(\dfrac{7}{10}x-\dfrac{3\times
25}{16}y^{4}\right)\omega_{X}\mod\left(\frac{\partial f}{\partial
x},\frac{\partial f}{\partial y}\right).$

Since $x^{i}v(f)=x^{i+1}f$, $i=1,2,3,4$ and $yv(f)=xyf$ hold, we can compute
the action of $Df$ on $x^{i+1}\omega_{X}$ and $xy\omega_{X}$ by using the
vector field $v$ above.

  2. $(ii)$

For $b(x,y)=y$, we have

$\displaystyle v=\frac{d_{1}}{5(4-25xy)}\frac{\partial}{\partial
x}+\frac{d_{2}}{5(4-25xy)}\frac{\partial}{\partial y},$

where $d_{1}=6xy-25x^{2}y^{2}$, $d_{2}=4y^{2}-25xy^{3}-5x^{3}$ and

$\displaystyle D(f(x,y)y\omega_{X})=\left(\dfrac{7}{10}y-\dfrac{3\times
25}{16}x^{4}\right)\omega_{X}\mod\left(\frac{\partial f}{\partial
x},\frac{\partial f}{\partial y}\right).$

Since the vector field $v$ above satisfies $v(f)=yf$, we also have
$y^{j}v(f)=y^{j+1}f$, $j=1,2,3$.

We can use these relations to compute the action of $Df$ on
$y^{j+1}\omega_{X}$, $j=1,2,3$. In this way, we obtain $\tau=10$ fundamental
relations.

Since the Milnor number $\mu$ is equal to 11, these 10 relations are not
enough to compute a connection matrix of the Gauss–Manin connection. We have
to compute the saturation.

Now recall the classical result on integral closure due to J. Briano̧n and H.
Skoda [38]. From the Briano̧n–Skoda theorem, we see that the function $f^{2}$
is in the ideal $J=\big{(}\frac{\partial f}{\partial x},\frac{\partial
f}{\partial y}\big{)}$. In [35], J. Scherk computed the following relation
explicitly and exploited it as the starting point for computing
$D(f^{2}\omega_{X})$ and $D(fD(f\omega_{X}))$:

$\displaystyle
25(4-25xy)f^{2}=\big{\\{}\big{(}20x-125x^{2}y\big{)}f+4x^{3}y^{2}-5xy^{5}-25x^{4}y^{3}\big{\\}}\frac{\partial
f}{\partial x}$
$\displaystyle\hphantom{25(4-25xy)f^{2}=\\{}{}+\big{\\{}\big{(}20y-125xy^{2}\big{)}f+6x^{2}y^{3}-25x^{3}y^{4}\big{\\}}\frac{\partial
f}{\partial y}.$

Here we propose a slightly different approach. By using an algorithm given in
[28], we can compute the following integral dependence relation

$\displaystyle 25(4-25xy)f^{2}=10x\bigg{(}\frac{\partial f}{\partial
x}\bigg{)}f\\!+\\!10y\bigg{(}\frac{\partial f}{\partial
y}\bigg{)}f\\!+\\!d_{2,0}\bigg{(}\frac{\partial f}{\partial
x}\bigg{)}^{2}\\!+\\!d_{1,1}\bigg{(}\frac{\partial f}{\partial
x}\bigg{)}\bigg{(}\frac{\partial f}{\partial
y}\bigg{)}\\!+\\!d_{0,2}\bigg{(}\frac{\partial f}{\partial y}\bigg{)}^{2},$

where

$\displaystyle d_{2,0}=2x^{2}-25x^{3}y-10y^{3},\qquad
d_{1,1}=11xy-50x^{2}y^{2},\qquad d_{0,2}=2y^{2}-25xy^{3}-10x^{3}.$

Compare to the relation used by Scherk, the integral dependence relation given
above represent much more precise relations between $f^{2}$,
$f\big{(}\frac{\partial f}{\partial x}\big{)}$, $f\big{(}\frac{\partial
f}{\partial y}\big{)}$, $\big{(}\frac{\partial f}{\partial x}\big{)}^{2}$,
$\big{(}\frac{\partial f}{\partial x}\big{)}\big{(}\frac{\partial f}{\partial
y}\big{)}$, $\big{(}\frac{\partial f}{\partial y}\big{)}^{2}$. Thanks to this
property, the use of the integral dependence relation, or the integral
equation leads an effective method for computing
$D\big{(}f^{2}\omega_{X}\big{)}$ and $D(fD(f\omega_{X}))$.

Note that in [28], we consider integral dependence relations in the context of
symbolic computation and introduced a concept of generalized integral
dependence relations. From this point of view relations obtained from non-
trivial logarithmic vector fields can be interpreted as generalized integral
dependence relations. These relations can also be computed by using the
algorithms described in [28].

Let $f(x)$ be a holomorphic function defined on $X\subset{\mathbb{C}}^{n}$.
Assume that the degree of integral equation, or the integral number of $f$
over the Jacobi ideal in the local ring ${\mathcal{O}}_{X,O}$ is equal to two.
Let

$f(x)^{2}+\sum_{i=1}^{n}a_{i}(x)f(x)\frac{\partial f}{\partial
x_{i}}(x)+\sum_{j\geq i}a_{i,j}(x)\frac{\partial f}{\partial
x_{i}}(x)\frac{\partial f}{\partial x_{j}}(x)=0$

be the integral equation of $f$. Then, from the Brieskorn formula, we have

$\displaystyle D(f(x)^{2}\omega_{X})$
$\displaystyle=-D\Bigg{\\{}\sum_{i=1}^{n}\Bigg{(}a_{i}(x)f(x)+\sum_{j\geq
i}a_{i,j}(x)\frac{\partial f}{\partial x_{j}}(x)\Bigg{)}\frac{\partial
f}{\partial x_{i}}(x)\omega_{X}\Bigg{\\}}$
$\displaystyle=-\Bigg{\\{}\sum_{i=1}^{n}\frac{\partial}{\partial
x_{i}}\Bigg{(}a_{i}(x)f(x)+\sum_{j\geq i}a_{i,j}(x)\frac{\partial f}{\partial
x_{j}}(x)\Bigg{)}\Bigg{\\}}\omega_{X}$

which is equal to

$-\Bigg{\\{}\sum_{i=1}^{n}a_{i}(x)\frac{\partial f}{\partial
x_{i}}(x)+\sum_{j\geq i}\frac{\partial a_{i,j}}{\partial
x_{i}}(x)\frac{\partial f}{\partial x_{j}}+R(x)\Bigg{\\}}\omega_{X},$

where

$R(x)=\bigg{(}\sum_{i=1}^{n}\frac{\partial a_{i}}{\partial
x_{i}}(x)\bigg{)}f(x)+\sum_{j\geq i}a_{i,j}(x)\frac{\partial^{2}f}{\partial
x_{i}\partial x_{j}}(x).$

If, there exist holomorphic functions $c_{i}(x)$, $i=1,2,\ldots,n$ such that

$R(x)=\sum_{i=1}^{n}c_{i}(x)\frac{\partial f}{\partial x_{i}}(x),$

then, we have for instance the following relation that can be used as a
starting point of the computation of a saturation

$D^{2}(f(x)^{2}\omega_{X})=-\Bigg{\\{}\sum_{i=1}^{n}\bigg{(}\frac{\partial
a_{i}}{\partial x_{i}}(x)+\frac{\partial c_{i}}{\partial
x_{i}}(x)\bigg{)}+\sum_{j\geq i}\frac{\partial^{2}a_{i,j}}{\partial
x_{j}\partial x_{i}}(x)\Bigg{\\}}\omega_{X}.$

Computing Gauss–Manin connections is a quite difficult problem [11, 14, 35,
36, 45]. We expect that the approach presented in this paper provides a method
to reduce difficulty to some extent.

### Acknowledgements

This work has been partly supported by JSPS Grant-in-Aid for Scientific
Research (C) (18K03320 and 18K03214).

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