# Bernstein-Sato polynomial and related invariants for meromorphic functions

Josep Àlvarez Montaner1 Departament de Matemàtiques and Institut de
Matemàtiques de la UPC-BarcelonaTech (IMTech)
Universitat Politècnica de Catalunya, Spain<EMAIL_ADDRESS>,  Manuel
González Villa2 Centro de investigación en Matemáticas
Apartado Postal 402
C.P. 36000, Guanajuato, GTO, México<EMAIL_ADDRESS>,  Edwin León-
Cardenal3 CONACYT–Centro de investigación en Matemáticas
Unidad Zacatecas, Av. Lasec Andador Galileo Galilei, Manzana 3 Lote 7
C.P. 98160, Zacatecas, ZAC, México<EMAIL_ADDRESS>and  Luis Núñez-
Betancourt4 Centro de investigación en Matemáticas
Apartado Postal 402
C.P. 36000, Guanajuato, GTO, México<EMAIL_ADDRESS>
###### Abstract.

We develop a theory of Bernstein-Sato polynomials for meromorphic functions.
As a first application we study the poles of Archimedian local zeta functions
for meromorphic germs. We also present a theory of multiplier ideals for
meromorphic functions from the analytic and algebraic point of view. It is
also shown that the jumping numbers of these multiplier ideals are related
with the roots of the Bernstein-Sato polynomials.

###### Key words and phrases:

Bernstein-Sato polynomial, Zeta Functions, Singularities

###### 2010 Mathematics Subject Classification:

Primary TBA; Secondary TBA.

1Partially supported by grant PID2019-103849GB-I00 funded by
MCIN/AEI/10.13039/501100011033.

2Partially supported by grant PID2020-114750GB-C33 funded by
MCIN/AEI/10.13039/501100011033.

3 The third author was partially supported by CONACyT Grant No. 286445.

4 The fourth author was partially supported by CONACyT Grant 284598 and
Cátedras Marcos Moshinsky.

###### Contents

  1. 1 Introduction
  2. 2 Background
  3. 3 Meromorphic Bernstein-Sato polynomial for polynomial rings
  4. 4 Meromorphic Bernstein-Sato polynomial for differentibly admissible algebras
  5. 5 Archimedean Local Zeta Functions
  6. 6 Multiplier ideals: analytic construction
  7. 7 Multiplier ideals: algebraic construction

##  1\. Introduction

The theory of Bernstein-Sato polynomials was introduced independently by
Bernstein [Ber72] to study the analytic continuation of the Archimedean zeta
function and Sato [SS72] in the context of prehomogeneous vector spaces.
Rapidly, Bernstein-Sato polynomials became an indispensable tool to study
invariants of singularities of holomorphic or regular functions. In
particular, the roots of the Bernstein-Sato polynomials are related to many
other invariants such as poles of zeta functions, eigenvalues of the monodromy
of the Milnor fiber, jumping numbers of multiplier ideals, spectral numbers
and F-thresholds, among others.

The aim of this paper is to develop a theory of Bernstein-Sato polynomials for
the case of meromorphic functions, and to relate it to other invariants of
such functions. This is inspired in previous work that has been done for
singularities of meromorphic functions. For instance, Arnold proposed the
study of singularities of meromorphic functions and began its classification
[Arn98]. He remarked that even simple examples provide new interesting
phenomena. Gussein-Zade, Luengo, and Melle [GZLMH98, GZLMH99a, GZLMH99b,
GZLMH01] pursued a systematic research of the topology and monodromy of germs
of complex meromorphic functions (see also works by Tibar and Siersma [Tib02,
ST04], Bodin, Pichon, and Seade [BP07, PS08, BPS09]), and by Raibaut [Rai12,
Rai13] studied motivic zeta functions and motivic Milnor fibers for
meromorphic germs, also considered by Libgober, Maxim, and the second author
[GVLM16, GVLM18]. Zúñiga-Galindo and the third author treated $p$-adic zeta
functions for Laurent polynomials [LCZG13]. Lemahieu and the second author
treated the case of topological zeta functions of meromorphic germs, and
proved a generalization of the monodromy conjecture in the two variable case
[GVL14]. Veys and Zúñiga-Galindo [VZG17] investigated meromorphic continuation
and poles of zeta functions and oscillatory integrals for meromorphic
functions defined over local fields of characteristic zero (see also [LC17,
BGZG18, BG20]). Nguyen and Takeuchi [NT19] introduced meromorphic nearby cycle
functors, and studied some of its applications to the monodromy of meromorphic
germs.

In this paper, we prove the existence of meromorphic Bernstein-Sato
polynomials for differentiably admissible $\mathbb{K}$-algebras, which is a
class of rings including polynomial and power series rings over $\mathbb{K}$
and rings of holomorphic functions in a neighbourhood of the origin in
$\mathbb{C}$ among other cases (see Section 2 for an overview).

######  Theorem A (Theorem 4.5).

Let $R$ be a differentiably admissible $\mathbb{K}$-algebra. Let $f,g\in R$ be
nonzero elements. There exists $b(s)\in\mathbb{K}[s]\setminus\\{0\\}$ and
$\delta(s)\in D_{R|\mathbb{K}}[s]$ such that

$\delta(s)\frac{f}{g}{\bf\left(\frac{f}{g}\right)^{s}}=b(s){\bf\left(\frac{f}{g}\right)^{s}}.$

More generally, we define a family of Bernstein-Sato polynomials
$b_{f/g}^{\alpha}(s)$ indexed by $\alpha\in\mathbb{R}_{\geq 0}$ in Proposition
4.2. We also provide a more elementary proof of the functional equation for
meromorphic functions of polynomial rings in Theorem 3.2. In the case of
holomorphic functions we provide a version of Kashiwara’s proof of the
rationality of the roots of the Bernstein-Sato polynomial [Kas77] and the
refinement given by Lichtin [Lic89].

######  Theorem B (Theorem 4.14, Corollary 4.15).

Let $f,g\in R$ be nonzero holomorphic functions and $\alpha\in\mathbb{R}_{\geq
0}$. The roots of $b_{f/g}^{\alpha}(s)$ are negative rational numbers.
Moreover, they are contained in the set

$\Bigg{\\{}\frac{k_{i}+1+\ell}{N_{f/g,i}}\;;\;\ell\geq 0,\;i\in
I_{0}\Bigg{\\}},$

where the integers $N_{f/g,i}$ and $k_{i}$ are extracted from the numerical
data of a log resolution of $f/g$.

As an application, we study meromorphic continuation of Archimedean local zeta
functions of meromorphic germs, extending the ideas of Bernstein [Ber72] in
the classical case. Our approach is complementary to the one done by Veys and
Zúñiga-Galindo [VZG17], where they used resolution of singularities following
the ideas of Bernstein and Gel’fand [BG69] and Atiyah [Ati70].

######  Theorem C (Theorem 5.4).

Let $f$ and $g$ be nonzero holomorphic functions and take
$\alpha=\operatorname{lct}_{0}g$. The local zeta function $Z_{\phi}(s,f/g)$
has a meromorphic continuation to the whole complex plane $\mathbb{C}$, and
its poles are contained in the set

$\Bigl{\\{}\zeta-k\alpha\ ;\ k\in\mathbb{Z}_{\geq 0}\Bigr{\\}}_{\zeta\text{
root of }b^{\alpha}_{f/g}(s)}\bigcup\Bigl{\\{}k\alpha-\xi\ ;\
k\in\mathbb{Z}_{\geq 0}\Bigr{\\}}_{\xi\text{ root of }b^{\alpha}_{g/f}(s)}.$

In particular, the poles of $Z_{\phi}(s,f/g)$ are rational numbers.

Finally, we study multiplier ideals of meromorphic functions from the analytic
and algebraic point of view.

######  Theorem D.

Let $f$ and $g$ be nonzero holomorphic functions. Then:

  * •

(Theorem 6.6) The set of jumping numbers of $f/g$ is a set of rational numbers
with no accumulation points.

  * •

(Theorem 6.7) Let $\lambda$ be a jumping number of $f/g$ such that
$\lambda\in(1-\operatorname{lct}_{0}(g),1]$. Then, $-\lambda$ is a root of the
Bernstein-Sato polynomial of $f/g$.

We also provide a weaker version of Skoda’s Theorem.

######  Proposition A.

Let $f$ and $g$ be nonzero holomorphic functions. Then:

  * •

(Proposition 6.5)
${\mathcal{J}}((\frac{f}{g})^{\lambda+\ell})=(\frac{f^{\ell}}{g^{\ell}}({\mathcal{J}}(\frac{f}{g})^{\lambda}))\bigcap
R$ for every $\ell\in\mathbb{N}.$ In particular, if $\lambda+1$ is a jumping
number, then $\lambda$ is a jumping number.

  * •

(Lemma 7.5) For every $\lambda\in\mathbb{R}_{>0}$, we have
${\mathcal{J}}(f^{\lambda})\subseteq{\mathcal{J}}((\frac{f}{g})^{\lambda})$.
In addition, we have ${\mathcal{J}}((\frac{f}{g})^{n})=(f^{n})$ for every
$n\in\mathbb{Z}_{>0}$.

Takeuchi has developed independently a theory of Bernstein-Sato polynomials
for meromorphic functions [Tak21]. Both approaches differ slightly (see Remark
4.8) but complement each other. Takeuchi is interested in the relation between
the roots of the Bernstein-Sato polynomial and the eigenvalues of the Milnor
monodromies. Meanwhile, we pay attention to the meromorphic continuation of
the Archimedean local zeta function and multiplier ideals.

##  2\. Background

###  2.1. Rings of differential operators and its modules

We start by recalling the basics on the theory of rings of differential
operators as introduced by Grothendieck [Gro67, §16.8].

######  Definition 2.1.

Let $R$ be a Noetherian ring containing a field of characteristic zero
$\mathbb{K}$. The ring of $\mathbb{K}$-linear differential operators of $R$ is
the subring $D_{R|\mathbb{K}}\subseteq\operatorname{Hom}_{\mathbb{K}}(R,R)$,
whose elements are defined inductively as follows.

  * •

Differential operators of order zero are defined by the multiplication by
elements of $R$, and so, $D^{0}_{R|\mathbb{K}}\cong R.$

  * •

An element $\delta\in\operatorname{Hom}_{\mathbb{Z}}(R,R)$ is an operator of
order less than or equal to $n$ if $[\delta,r]=\delta r-r\delta$ is an
operator of order less than or equal to $n-1.$

We denote by $D^{n}_{R|\mathbb{K}}$ the set of all differential operators of
order less than or equal to $n$. We have a filtration
$D^{0}_{R|\mathbb{K}}\subseteq D^{1}_{R|\mathbb{K}}\subseteq\cdots$ such that
$D^{m}_{R|\mathbb{K}}D^{n}_{R|\mathbb{K}}\subseteq D^{m+n}_{R|\mathbb{K}}$. We
denote this filtration by $D^{\bullet}_{R|\mathbb{K}}$. The ring of
differential operators is defined as

$D_{R|\mathbb{K}}=\bigcup_{n\in\mathbb{N}}D^{n}_{R|\mathbb{K}}.$

We note that in general
$D^{1}_{R|\mathbb{K}}\cong\operatorname{Der}_{R|\mathbb{K}}\bigoplus R$, but
$D_{R|\mathbb{K}}$ is not always generated as $R$-algebra by derivations.

###  2.2. Holonomic $D$-modules for polynomial rings

######  Definition 2.2.

Let $R=\mathbb{C}[x_{1},\ldots,x_{d}]$. We define the Bernstein filtration of
$R$, $\mathcal{B}^{\bullet}_{R|\mathbb{C}}$ by

$\mathcal{B}^{i}_{R|\mathbb{C}}=\mathbb{C}\\{x^{\alpha}\delta^{\beta}\;;\;|\alpha|+|\beta|\leq
i\\}.$

We note that

  * •

$\dim_{\mathbb{C}}\mathcal{B}^{i}_{R|\mathbb{C}}=\binom{n+i}{i}<\infty$,

  * •

$D_{R|\mathbb{C}}=\bigcup_{i\in\mathbb{N}}\mathcal{B}^{i}_{R|\mathbb{C}}$, and

  * •

$\mathcal{B}^{n}_{R|\mathbb{C}}\mathcal{B}^{j}_{R|\mathbb{C}}=\mathcal{B}^{j}_{R|\mathbb{C}}$.

We observe that
$\operatorname{gr}_{\mathcal{B}^{\bullet}_{R|\mathbb{C}}}(D_{R|\mathbb{C}})$
is a commutative ring isomorphic to
$\mathbb{C}[x_{1},\ldots,x_{d},y_{1},\ldots,y_{d}]$.

######  Definition 2.3.

Given a left (right) $D_{R|\mathbb{C}}$-module, $M$, we say that a filtration
$\Gamma^{\bullet}$ of $\mathbb{C}$-vector spaces is
$\mathcal{B}^{\bullet}_{R|\mathbb{C}}$-compatible if

  * •

$\dim_{\mathbb{C}}\Gamma^{i}<\infty$,

  * •

$M=\bigcup_{i\in\mathbb{N}}\Gamma^{i}$,

  * •

$\mathcal{B}^{i}_{R|\mathbb{C}}\Gamma^{j}\subseteq\Gamma^{j}$
($\Gamma^{j}\mathcal{B}^{i}_{R|\mathbb{C}}\subseteq\Gamma^{j}$).

We say that $\Gamma^{\bullet}$ is a good filtration if
$\operatorname{gr}_{\Gamma^{\bullet}}(M)$ is finitely generated as a
$\operatorname{gr}_{\mathcal{B}^{\bullet}_{R|\mathbb{C}}}(D_{R|\mathbb{C}})$-module.

If $\Gamma^{\bullet}$ is a good filtration for $M$, we have that
$\dim_{\mathbb{C}}\Gamma^{n}$ is a polynomial function on $n$ of degree equal
to the Krull dimension of $\operatorname{gr}_{\Gamma^{\bullet}}(M)$ by
Hilbert-Samuel theory. This degree does not depend of the choice of such good
filtration, and it is called $\dim_{D_{R|\mathbb{C}}}(M)$. We note that every
finitely generated $D_{R|\mathbb{C}}$-module has a good filtration.

######  Theorem 2.4 (Bernstein’s Inequality).

Let $M$ be a finitely generated $D_{R|\mathbb{C}}$-module. Then,

$d\leq\dim_{D_{R|\mathbb{C}}}(M)\leq 2d.$

######  Definition 2.5.

A $D_{R|\mathbb{C}}$-module $M$, is holonomic if either $M=0$ or
$\dim_{D_{R|\mathbb{C}}}(M)=d$.

We observe that a $D_{R|\mathbb{C}}$-module $M$, with a good filtration
$\Gamma^{\bullet}$ is holonomic if and only if there exists a polynomial $q$
in one variable of degree $d$, such that $\dim_{\mathbb{C}}\Gamma^{n}\leq
q(n)$. We recall that every holonomic $D_{R|\mathbb{C}}$ has finite length.

###  2.3. Rings differentiably admissible

We now introduce a family of $\mathbb{K}$-algebras whose rings of differential
operators satisfy good properties.

######  Definition 2.6.

Let $\mathbb{K}$ be a field of characteristic zero. Let $R$ be a Noetherian
regular $\mathbb{K}$-algebra of dimension $d$. We say that $R$ is
differentiably admissible if

  1. (1)

$\dim(R_{\mathfrak{m}})=d$ for every maximal ideal $\mathfrak{m}\subseteq R$,

  2. (2)

$R/\mathfrak{m}$ is an algebraic extension of $\mathbb{K}$ for every maximal
ideal $\mathfrak{m}\subseteq R$, and

  3. (3)

$\operatorname{Der}_{R|\mathbb{K}}$ is a projective $R$-module of rank $d$
such that the natural map

$R_{\mathfrak{m}}\otimes_{R}\operatorname{Der}_{R|\mathbb{K}}\to\operatorname{Der}_{R_{\mathfrak{m}}|\mathbb{K}}$

is an isomorphism.

This class of algebras were introduced by the third-named author [NB13] to
obtain the existence of the Bernstein-Sato polynomial for algebras that do not
have global coordinates. This later case was already studied by Narváez
Macarro and Mebkhout [MNM91]. Examples of differentiably admissible algebras
include polynomial rings over $\mathbb{K}$, power series rings over
$\mathbb{K}$, the ring of convergent power series in a neighborhood of the
origin over $\mathbb{C}$, Tate and Dwork-Monsky-Washnitzer
$\mathbb{K}$-algebras [MNM91], the localization of complete regular rings of
mixed characteristic at the uniformizer [NB13, Lyu00], and the localization of
complete local domains of equal-characteristic zero at certain elements
[Put18].

######  Theorem 2.7 ([NB13, Section 2]).

Let $\mathbb{K}$ be a field of characteristic zero. Let $R$ be a
differentiably admissible $\mathbb{K}$-algebra of dimension $d$. Then,

  1. (1)

$D^{n}_{R|\mathbb{K}}=(\operatorname{Der}_{R|\mathbb{K}}+R)^{n}$, and

  2. (2)

$D_{R|\mathbb{K}}\cong R\langle\operatorname{Der}_{A|\mathbb{K}}\rangle$.

  3. (3)

$D_{R|\mathbb{K}}$ is left and right Noetherian;

  4. (4)

$\operatorname{gr}_{D^{\bullet}_{R|\mathbb{K}}}(D_{R|\mathbb{K}})$ is a
regular ring of pure graded dimension $2d$;

  5. (5)

$\operatorname{gl.dim}(D_{R|\mathbb{K}})=d$.

In particular, $D_{R|\mathbb{K}}$ is an $R$-algebra of differentiable type
with the order filtration.

######  Theorem 2.8 ([NB13, Section 2]).

Let $\mathbb{K}$ be a field of characteristic zero, and
$\mathbb{K}(s)=\operatorname{Frac}(\mathbb{K}[s])$. Let $R$ be a
differentiably admissible $\mathbb{K}$-algebra of dimension $d$, and
$R(s)=R\otimes_{\mathbb{K}}\mathbb{K}(s)$. Then,
$D(s)=D_{R|\mathbb{K}}\otimes_{\mathbb{K}}\mathbb{K}(s)$ is a $R(s)$-algebra
of differentiable type with the order filtration.

######  Definition 2.9.

Let $A$ be an $R$-algebra with $1$. By a filtration
$F^{\bullet}=\\{F^{i}\\}_{i\in\mathbb{N}}$ on $A$ we mean an ascending
filtration of finitely generated $R$-modules $F^{i}$ such that
$A=\bigcup_{i\in\mathbb{N}}F^{i}$ and $F^{i}F^{j}\subseteq F^{i+j}$. In this
case we say that $(A,F^{\bullet})$ is filtered. We say that a left $A$-module
$M$ with a filtration $\Gamma^{\bullet}=\\{\Gamma^{i}\\}_{i\in\mathbb{N}}$ is
$F^{\bullet}$-compatible if

  1. (1)

$\Gamma^{i}$ is finitely generated,

  2. (2)

$M=\bigcup_{i\in\mathbb{N}}\Gamma^{i}$,

  3. (3)

$F^{i}\Gamma^{j}\subseteq\Gamma^{j}$.

We say that $\Gamma^{\bullet}$ is a good filtration if
$\operatorname{gr}_{\Gamma^{\bullet}}(M)$ is finitely generated as a
$\operatorname{gr}_{F^{\bullet}}(A)$-module. We also consider the analogue
definitions for right $A$-modules.

######  Definition 2.10.

We say that a filtered $R$-algebra $(A,F^{\bullet})$ is an algebra of
differentiable type if $\operatorname{gr}_{F^{\bullet}}(A)$ is a commutative
Noetherian ring with $1$ which is regular with pure graded dimension.

######  Remark 2.11.

Let $A$ be an $R$-algebra of differentiable type. We note that every two good
filtrations, $\Gamma^{\bullet}$ and $\widetilde{\Gamma}^{\bullet}$, of $M$ are
shift cofinal. Specifically, there exists $a\in\mathbb{N}$ such that

$\Gamma^{i-a}\subseteq\widetilde{\Gamma^{i}}\subseteq\Gamma^{i+a}.$

Furthermore, $M$ is finitely generated as $A$-module if and only if $M$ has a
good filtration.

######  Definition 2.12.

Let $A$ be a filtered $R$-algebra, and $M$ be a left $A$-module. The dimension
of $M$ is defined by

$\dim_{A}(M)=\dim_{\operatorname{gr}_{F^{\bullet}}(A)}\operatorname{gr}_{\Gamma^{\bullet}}(M),$

where $\Gamma^{\bullet}$ is a good filtration of $M$.

######  Definition 2.13.

Let $A$ be an algebra of differentiable type. Let $M\neq 0$ be a finitely
generated $A$-module. We define

$\operatorname{grade}_{A}(M)=\hbox{inf}\\{j\ |\
\operatorname{Ext}^{j}_{A}(M,A)\neq 0\\}.$

We note that
$\operatorname{grade}_{A}(M)\leq\operatorname{gl.dim}(A)=\dim(A)-\operatorname{gl.dim}(A).$

######  Proposition 2.14 ([Bjö79, Chapter 2., Theorem 7.1]).

Let $A$ be an algebra of differentiable type, and let $M\neq 0$ be a finitely
generated $A$-module. Then,

$\dim_{A}(M)+\operatorname{grade}_{A}(M)=\dim(A).$

In particular, $\dim_{A}(M)\geq\dim(A)-\operatorname{gl.dim}(A).$

######  Definition 2.15.

Let $A$ be a $\mathbb{K}$-algebra of differentiable type. Let $M$ be a
finitely generated left (right) $A$-module. We say that $M$ is in the _left
(right) Bernstein class_ if either $M=0$ or
$\dim_{A}(M)=\dim(A)-\operatorname{gl.dim}(A)$.

This class of Bernstein modules is an analogue of the class of holonomic
modules. In particular, it is closed under submodules, quotients, extensions,
and localizations [MNM91, Proposition 1.2.7]).

Let $A$ be a $\mathbb{K}$-algebra of diferentiable type and global dimension
$d$, and $M$ be a finitely generated $A$-module. If $M$ is in the Bernstein
class of $A$, then $\operatorname{Ext}^{i}_{A}(M,A)\neq 0$ if and only if
$i=d$ [Bjö79]. Then, the functor that sends $M$ to
$\operatorname{Ext}^{d}_{A}(M,A)$ is an exact contravariant functor that
interchanges the left and the right Bernstein class. Furthermore,
$M\cong\operatorname{Ext}^{d}_{A}(\operatorname{Ext}^{d}_{A}(M,A),A)$ for
modules in the Bernstein class. Since $A$ is left and right Noetherian, the
modules in the Bernstein class are both Noetherian and Artinian. We conclude
that the modules in the Bernstein class have finite length as $A$-modules
[MNM91, Proposition 1.2.5])

###  2.4. Log-resolution of meromorphic functions

Let $R$ be a differentiably admissible $\mathbb{K}$-algebra. Given nonzero
elements $f,g\in R$ we are interested in the function $f/g$. Most likely
$f,g:(\mathbb{C}^{n},0)\longrightarrow(\mathbb{C},0)$ are germs of holomorphic
functions and we refer to
$f/g:(\mathbb{C}^{n},0)\longrightarrow(\mathbb{C},0)$ as a germ of a
meromorphic function. Taking local coordinates we will assume $f,g\in
R=\mathbb{C}\\{x_{1},\dots,x_{n}\\}$. Recall that $f/g$ and
$f^{\prime}/g^{\prime}$ define the same germ if there exist a unit
$u\in\mathcal{O}_{\mathbb{C}^{n},0}$ such that $f=uf^{\prime}$ and
$g=ug^{\prime}$.

######  Definition 2.16.

Let $X$ be a $n$-dimensional smooth analytic manifold, $U$ a neighborhood of
$0\in\mathbb{C}^{n}$ and $\pi:X\to U$ a proper analytic map. We say that $\pi$
is log resolution of the meromorphic germ $f/g$ if:

  * •

$\pi$ is a log resolution of the hypersurface
$H=\\{f_{|_{U}}=0\\}\cup\\{g_{|_{U}}=0\\}$, i.e. $\pi$ is an isomorphism
outside a proper analytic subspace in $U$;

  * •

there is a normal crossing divisor $F$ on $X$ such that
$\pi^{-1}(H)=\mathcal{O}_{X}(-F)$;

  * •

the lifting $\tilde{f}/\tilde{g}=(f/g)\circ\pi=\frac{f\circ\pi}{g\circ\pi}$
defines a holomorphic map $\tilde{f}/\tilde{g}:X\to\mathbb{P}^{1}$.

One can obtain a log resolution $\pi$ of $f/g$ from a log resolution
$\pi^{\prime}$ of $f\cdot g$ by blowing up along the intersections of
irreducible components of $\pi^{\prime-1}\\{f\cdot g=0\\}$ until the
irreducible components of the strict transform in $\pi^{-1}\\{f=0\\}$ and
$\pi^{-1}\\{g=0\\}$ are separated by a dicritical component, i.e. an
exceptional divisor $E$ of $\pi$ for which
$(\tilde{f}/\tilde{g})_{|_{E}}:E\to\mathbb{P}^{1}$ is a surjective map.

Let $\\{E_{i}\\}_{i\in I}$ be the irreducible components of $F$. We denote the
relative canonical divisor, defined by the Jacobian determinant of $\pi$, as
the divisor

$K_{\pi}=\sum k_{i}E_{i}.$

We also denote the strict transforms of $f$ and $g$ as

$\tilde{f}=\pi^{*}f=\sum N_{f,i}E_{i},\quad\tilde{g}=\pi^{*}g=\sum
N_{g,i}E_{i}.$

Moreover, we define

$N_{f/g,i}=N_{f,i}-N_{g,i}.$

Notice that $E_{i}$ is a dicritical component if and only if $N_{f/g,i}=0$. It
is then natural to associate with the meromorphic germ $f/g$ the divisor

$\tilde{F}=\sum_{i\in I}N_{f/g,i}E_{i}.$

For $i\in I$ we write $i\in I_{0}$ if $N_{f,i}>N_{g,i}$, $i\in I_{\infty}$ if
$N_{g,i}>N_{f,i}$, and $i\in I_{d}$ if $N_{f_{i}}=N_{g,i}$. We have the
decomposition $\tilde{F}=\tilde{F}_{0}+\tilde{F}_{\infty}+\tilde{F}_{d}$ where

$\tilde{F}_{0}=\sum_{i\in I_{0}}N_{f/g,i}E_{i},\hskip
14.22636pt\tilde{F}_{\infty}=\sum_{i\in I_{\infty}}N_{f/g,i}E_{i},\hskip
14.22636pt\textrm{and}\hskip 14.22636pt\tilde{F}_{d}=\sum_{i\in
I_{d}}N_{f/g,i}E_{i},$

with $\tilde{F}_{d}=0$ by definition.

######  Remark 2.17.

Around a given point $p\in X$ we may consider a local system of coordinates
$(z_{1},\dots,z_{n})$ such that the components $E_{i}$ containing $p$ are
given by the equations $z_{i}=0$. Assume that $p\in E_{i}$, for $i=1,\dots,m$,
then we have the local equations:

$\tilde{f}=uz_{1}^{N_{f,1}}\cdots
z_{m}^{N_{f,r}},\quad\text{and}\quad\tilde{g}=vz_{1}^{N_{g,1}}\cdots
z_{m}^{N_{g},r},$

where $u,v\in{\mathcal{O}}_{X,p}$ are units. Moreover, the local equations of
the relative canonical divisor are

$\tilde{\omega}=\pi^{\ast}(dx_{1}\wedge\cdots\wedge
dx_{n})=w(z_{1}^{k_{1}}\cdots z_{m}^{k_{m}})\,dz_{1}\wedge\cdots\wedge
dz_{n},$

where $w\in{\mathcal{O}}_{X,p}$ is a unit as well. Furthermore, applying some
extra blow-ups if necessary, we may assume that either $\tilde{f}$ divides
$\tilde{g}$ or the other way around, so we have either

$\frac{\tilde{f}}{\tilde{g}}=\frac{u}{v}\;z_{1}^{N_{f/g,1}}\cdots
z_{m}^{N_{f/g,m}}\hskip 14.22636pt\textrm{or}\hskip
14.22636pt\frac{\tilde{f}}{\tilde{g}}=\frac{u}{v}\;\frac{1}{z_{1}^{|N_{f/g,1}|}\cdots
z_{m}^{|N_{f/g,m}|}}.$

##  3\. Meromorphic Bernstein-Sato polynomial for polynomial rings

Our first goal is to introduce the theory of Bernstein-Sato polynomials for
quotients of polynomials. We single out this case for the convenience of the
reader since it follows the same lines of reasoning as in the classical case.

######  Definition 3.1.

Let $f,g\in R=\mathbb{C}[x_{1},\ldots,x_{d}]$ be nonzero elements. We set
$M_{f/g}=R_{fg}{\bf\left(\frac{f}{g}\right)^{s}}$. This free module has an
structure of $D_{R|\mathbb{C}}[s]$-modulo given by

$\partial\cdot\frac{h}{f^{a}g^{b}}{\bf\left(\frac{f}{g}\right)^{s}}=\partial\left(\frac{h}{f^{a}g^{b}}\right){\bf\left(\frac{f}{g}\right)^{s}}+s\frac{h}{f^{a+1}g^{b-1}}\partial\left(\frac{f}{g}\right){\bf\left(\frac{f}{g}\right)^{s}}.$

where $\partial$ is a $\mathbb{C}$-linear derivative on $R$.

######  Theorem 3.2.

Let $f,g\in R=\mathbb{C}[x_{1},\ldots,x_{d}]$ be a nonzero elements. There
exists $b(s)\in\mathbb{C}[s]\setminus\\{0\\}$ and $\delta(s)\in
D_{R|\mathbb{C}}[s]$ such that

$\delta(s)\frac{f}{g}{\bf\left(\frac{f}{g}\right)^{s}}=b(s){\bf\left(\frac{f}{g}\right)^{s}}.$

###### Proof.

Let $R(s)=R\otimes_{\mathbb{C}[s]}\mathbb{C}(s)$ and
$D(s)=D_{R(s)|\mathbb{C}(s)}$, where $\mathbb{C}(s)$ denotes the fraction
field of $\mathbb{C}[s]$. Let $M=M_{f/g}\otimes_{\mathbb{C}[s]}\mathbb{C}(s)$.
We claim that $M$ is a holonomic $D(s)$-module. Let $\theta=\deg(f)+\deg(g)$.
We set a filtration of finite dimensional $\mathbb{C}$-vector spaces.

$\Gamma^{n}=\frac{1}{f^{n}g^{n}}\left\\{h{\bf\left(\frac{f}{g}\right)^{s}}\;|\;\deg(h)\leq(\theta+1)n\right\\}.$

We have that $x_{i}\Gamma^{n}\subseteq\Gamma^{n+1}$. We now observe that
$\partial_{i}\Gamma^{n}\subseteq\Gamma^{n+1}$. Given
$\frac{h}{f^{n}g^{n}}{\bf\left(\frac{f}{g}\right)^{s}}\in\Gamma^{n},$ we have
that

$\displaystyle\partial_{i}\frac{h}{f^{n}g^{n}}{\bf\left(\frac{f}{g}\right)^{s}}$
$\displaystyle=\frac{f^{n}g^{n}\delta_{i}(h)-nhf^{n-1}g^{n-1}\partial_{i}(fg)}{f^{2n}g^{2n}}{\bf\left(\frac{f}{g}\right)^{s}}+\frac{sh}{f^{n+1}g^{n-1}}\left(\frac{g\delta_{i}(f)-f\delta_{i}(g)}{g^{2}}\right){\bf\left(\frac{f}{g}\right)^{s}}$
$\displaystyle=\frac{fg\delta_{i}(h)-nh\partial_{i}(fg)}{f^{n+1}g^{n+1}}{\bf\left(\frac{f}{g}\right)^{s}}+\frac{sh(g\delta_{i}(f)-f\delta_{i}(g))}{f^{n+1}g^{n+1}}{\bf\left(\frac{f}{g}\right)^{s}}.$

By considering the degrees of the polynomials appearing in the numerators, we
have that $\partial_{i}\Gamma^{n}\subseteq\Gamma_{n+1}$. Then,
$\Gamma^{\bullet}$ is a filtration compatible with the Bernstein filtration.
We observe that
$\dim_{\mathbb{C}(s)}\Gamma^{n}\leq\dim_{\mathbb{C}}[R]_{\leq(\theta+1)n}$,
where the latter is a polynomial on $n$ of degree $d$. Then, $M$ is a
holonomic $D(s)$-module.

Since $M$ is holonomic, it has finite length, and the sequence of
$D(s)$-submodules

$D(s)\frac{1}{fg}{\bf\left(\frac{f}{g}\right)^{s}}\supseteq
D(s)\frac{1}{f^{2}g}{\bf\left(\frac{f}{g}\right)^{s}}\supseteq
D(s)\frac{1}{f^{3}g}{\bf\left(\frac{f}{g}\right)^{s}}\supseteq\ldots$

stabilizes. There exists $m\in\mathbb{N}$ such that

$D(s)\frac{1}{f^{m-1}g}{\bf\left(\frac{f}{g}\right)^{s}}=D(s)\frac{1}{f^{m}g}{\bf\left(\frac{f}{g}\right)^{s}}.$

Thus, there exists $\delta(s)\in D(s)$ such that

$\delta(s)\left(\frac{1}{f^{m-1}g}\right){\bf\left(\frac{f}{g}\right)^{s}}=\left(\frac{1}{f^{m}g}\right){\bf\left(\frac{f}{g}\right)^{s}}.$

Then, there exists $b(s)\in\mathbb{C}[s]\setminus\\{0\\}$ such that
$\widetilde{\delta}(s)=b(s)\delta(s)\in D_{R|\mathbb{C}}[s]$. Hence,

(3.0.1)
$\widetilde{\delta}(s)\left(\frac{1}{f^{m-1}g}\right){\bf\left(\frac{f}{g}\right)^{s}}=b(s)\frac{1}{f^{m}g}{\bf\left(\frac{f}{g}\right)^{s}}.$

We have an isomorphism of $D_{R[s]|\mathbb{C}[s]}$-modules $\psi:M_{f/g}\to
M_{f/g}$ defined by

$\frac{p(s)h}{f^{\alpha}g^{\beta}}{\bf\left(\frac{f}{g}\right)^{s}}\mapsto\frac{p(s-m)h}{f^{\alpha+m}g^{\beta-m}}{\bf\left(\frac{f}{g}\right)^{s}}.$

After applying $\psi$ to Equation 3.0.1, we obtain that

$\widetilde{\delta}(s-m)\frac{f}{g}{\bf\left(\frac{f}{g}\right)^{s}}=b(s-m)\frac{1}{g}{\bf\left(\frac{f}{g}\right)^{s}}.$

Finally multiplying by $g$, we obtain the desired equation. ∎

##  4\. Meromorphic Bernstein-Sato polynomial for differentibly admissible
algebras

In this Section we will generalize the Bernstein-Sato theory to the case of
differentibly admissible algebras. As in the polynomial case, the functional
equation takes place in the following module.

######  Definition 4.1.

Let $R$ be a differentiably admissible $\mathbb{K}$-algebra. Let $f,g\in R$ be
nonzero elements, take $\alpha\in\mathbb{R}_{\geq 0}$ and set
$\mathscr{M}^{\alpha}_{f/g}[s]=R_{fg}[s]{\bf\frac{f^{s}}{g^{s+\alpha}}}$. This
free module has an structure of $D_{R|\mathbb{K}}[s]$-module given by

$\partial\cdot\frac{h(s)}{f^{a}g^{b}}{\bf\frac{f^{s}}{g^{s+\alpha}}}=\partial\left(\frac{h(s)}{f^{a}g^{b}}\right){\bf\frac{f^{s}}{g^{s+\alpha}}}+s\frac{\partial(f)h(s)}{f^{a+1}g^{b}}{\bf\frac{f^{s}}{g^{s+\alpha}}}-(s+\alpha)\frac{h(s)\partial(g)}{f^{a}g^{b+1}}{\bf\frac{f^{s}}{g^{s+\alpha}}},$

where $\partial$ is a $\mathbb{K}$-linear derivative on $R$. If $\alpha=0$, we
just write $\mathscr{M}_{f/g}[s]=R_{fg}[s]{\bf\left(\frac{f}{g}\right)^{s}}$.

######  Proposition 4.2.

Let $R$ be a differentiably admissible $\mathbb{K}$-algebra. Let $f,g\in R$ be
nonzero elements and $\alpha\in\mathbb{R}_{\geq 0}$. There exists
$b(s)\in\mathbb{K}[s]\setminus\\{0\\}$ and $\delta(s)\in D_{R|\mathbb{K}}[s]$
such that

$\delta(s)f{\bf\frac{f^{s}}{g^{s+\alpha}}}=b(s){\bf\frac{f^{s}}{g^{s+\alpha}}}.$

###### Proof.

We have that the global dimensions of $D_{R(s)|\mathbb{K}(s)}$ and
$D_{R|\mathbb{K}}$ are equal to $d=\dim(R)$ [MNM91, Proposition 2.2.3]. For
the sake of simplicity we denote $D=D_{R|\mathbb{K}}$,
$D(s)=D_{R(s)|\mathbb{K}(s)}$ $\mathscr{M}[s]=\mathscr{M}^{\alpha}_{f/g}[s]$,
and $\mathscr{M}(s)=\mathscr{M}[s]\otimes_{\mathbb{K}}\mathbb{K}(s)$. We also
consider $R_{fg}$ with the usual action of $D_{R|\mathbb{K}}$ and we set
$R_{fg}(s)=R_{fg}[s]\otimes_{\mathbb{K}[s]}\mathbb{K}(s)$, and
$D_{fg}(s)=R_{fg}\otimes_{R}D(s)$. We have a chain of isomorphisms

$\displaystyle\operatorname{Ext}^{i}_{D_{f}(s)}(\mathscr{M}(s),D_{fg}(s))$
$\displaystyle\cong\operatorname{Ext}^{i}_{D_{fg}(s)}(R_{fg}(s)\otimes_{R_{fg}(s)}\mathscr{M}(s),D_{fg}(s))$
$\displaystyle\cong\operatorname{Ext}^{i}_{D_{fg}(s)}(R_{fg}(s),\operatorname{Hom}_{R_{fg}(s)}(\mathscr{M}(s),D_{fg}(s)))$
$\displaystyle\cong\operatorname{Ext}^{i}_{D_{fg}(s)}(R_{fg}(s),D_{fg}(s))\otimes_{D_{fg}(s)}\operatorname{Hom}_{R_{fg}(s)}(\mathscr{M}(s),D_{fg}(s)))$
$\displaystyle\cong\operatorname{Ext}^{i}_{D_{fg}(s)}(R_{fg},D_{fg})\otimes_{D_{fg}}\operatorname{Hom}_{R_{fg}(s)}(\mathscr{M}(s),D_{fg}(s)))$
$\displaystyle\cong(\operatorname{Ext}^{i}_{D}(R,D)\otimes_{R}R_{fg})\otimes_{D_{fg}}\operatorname{Hom}_{R_{fg}(s)}(\mathscr{M}(s),D_{fg}(s))).$

Since $\operatorname{Ext}^{i}_{D}(R,D)\otimes_{R}R_{fg}=0$ for $i\neq d$, we
conclude that
$\operatorname{Ext}^{i}_{D_{fg}(s)}(\mathscr{M}(s),D_{fg}(s))\neq 0$ for
$i\neq d$. Then, $\mathscr{M}(s)$ has a $D(s)$-submodule $N$ in the Bernstein
class of $D(s)$ such that $N_{fg}=\mathscr{M}(s)$ [MNM91, Proposition 1.2.7
and Proof of Theorem 3.1.1]. Then, there exists $\ell\in\mathbb{N}$ such that
$f^{\ell}g^{\ell}{\bf\frac{f^{s}}{g^{s+\alpha}}}\in N$. Since $N$ has finite
length as $D_{A(s)|\mathbb{K}(s)}$-module the chain

$D(s)f^{\ell}g^{\ell}{\bf\frac{f^{s}}{g^{s+\alpha}}}\supseteq
D(s)f^{\ell+1}g^{\ell}{\bf\frac{f^{s}}{g^{s+\alpha}}}\supseteq
D(s)f^{\ell+1}g^{\ell}{\bf\frac{f^{s}}{g^{s+\alpha}}}\supseteq\ldots$

stabilizes. Then, there exists $m\in\mathbb{N}$ and a differential operator
$\delta(s)\in D_{A(s)|\mathbb{K}(s)}$ such that

$\delta(s)f^{\ell+m+1}{\bf\frac{f^{s}}{g^{s+\alpha}}}=f^{\ell+m}{\bf\frac{f^{s}}{g^{s+\alpha}}}.$

After clearing denominators and applying a shifting, there exists
$\widetilde{\delta}(s)\in D_{A|\mathbb{K}}[s]$ such that

$\widetilde{\delta}(s)f{\bf f^{s}}={\bf f^{s}}.$

∎

######  Definition 4.3.

Let $R$ be a differentiably admissible $\mathbb{K}$-algebra. Let $f,g\in R$ be
nonzero elements and $\alpha\in\mathbb{R}_{\geq 0}$. The Bernstein-Sato
polynomial $b_{f/g}^{\alpha}(s)\in\mathbb{K}[s]$ of order $\alpha$ of the
meromorphic function $f/g$ is the monic polynomial of smallest degree
satisfying the functional equation in Proposition 4.2.

######  Proposition 4.4.

Let $R$ be a differentiably admissible $\mathbb{K}$-algebra. Let $f,g\in R$ be
nonzero elements and $\alpha\in\mathbb{R}_{\geq 0}$. The following are
equivalent:

  1. (1)

The Bernstein-Sato polynomial of $f/g$ of order $\alpha$;

  2. (2)

The minimal polynomial of the action of $s$ on
$\displaystyle\frac{D_{R|\mathbb{K}}[s]{\bf\frac{f^{s}}{g^{s+\alpha}}}}{D_{R|\mathbb{K}}[s]f{\bf\frac{f^{s}}{g^{s+\alpha}}}}$;

  3. (3)

The monic element of smallest degree in
$\mathbb{K}[s]\cap(\mathrm{Ann}_{D_{R|\mathbb{K}}[s]}({\bf\frac{f^{s}}{g^{s+\alpha}}})+D_{R|\mathbb{K}}[s]f)$.

###### Proof.

The equivalence between the first two follows from the definition. For the
equivalence between the second and the third, we observe that

$\displaystyle\frac{D_{R|\mathbb{K}}[s]{\bf\frac{f^{s}}{g^{s+\alpha}}}}{D_{R|\mathbb{K}}[s]f{\bf\frac{f^{s}}{g^{s+\alpha}}}}$
$\displaystyle\cong\mathrm{coker}\left(\frac{D_{R|\mathbb{K}}[s]}{\mathrm{Ann}_{D_{R|\mathbb{K}}[s]}({\bf
f^{s}})}\xrightarrow{\cdot
f}\frac{D_{R|\mathbb{K}}[s]}{\mathrm{Ann}_{D_{R|\mathbb{K}}[s]}({\bf\frac{f^{s}}{g^{s+\alpha}}})}\right)$
$\displaystyle\cong\frac{D_{R|\mathbb{K}}[s]}{\mathrm{Ann}_{D_{R|\mathbb{K}}[s]}({\bf\frac{f^{s}}{g^{s+\alpha}}})+D_{R|\mathbb{K}}[s]f}.$

∎

We give a $D_{R[t]|\mathbb{K}}$-module structure on
$\mathscr{M}^{\alpha}_{f/g}[s]$ where the new variable $t$ acts as
multiplication by $f$. We define

$t\cdot\frac{h(s)}{f^{a}g^{b}}{\bf\frac{f^{s}}{g^{s+\alpha}}}=\frac{h(s+1)}{f^{a}g^{b}}f{\bf\frac{f^{s}}{g^{s+\alpha}}}$

$\partial_{t}\cdot\frac{h(s)}{f^{a}g^{b}}{\bf\frac{f^{s}}{g^{s+\alpha}}}=\frac{h(s-1)}{f^{a}g^{b}}\frac{1}{f}{\bf\frac{f^{s}}{g^{s+\alpha}}}$

A simple computation shows that $\partial_{t}t-t\partial_{t}=1$ and that
$-\partial_{t}t$ acts as multiplication by $s$. Moreover $ts-st=t$ and we may
consider the ring $D_{R|\mathbb{K}}\langle s,t\rangle\subseteq
D_{R[t]|\mathbb{K}}$. Denote for simplicity

$\mathscr{N}_{f/g}^{\alpha}=D_{R|\mathbb{K}}[s]{\bf\frac{f^{s}}{g^{s+\alpha}}}\subseteq\mathscr{M}^{\alpha}_{f/g}[s]$

We have that $\mathscr{N}^{\alpha}_{f/g}$ is indeed a $D_{R|\mathbb{K}}\langle
s,t\rangle$-module. As in the classical case, we may view the Bernstein-Sato
polynomial of $f/g$ of order $\alpha$ as the minimal polynomial of the action
of $-\partial_{t}t=s$ on

$\displaystyle\frac{\mathscr{N}^{\alpha}_{f/g}}{t\;\mathscr{N}^{\alpha}_{f/g}}$

Building upon the previous construction we deduce a Bernstein-Sato polynomial
of a meromorphic function that mimics the classical case.

######  Theorem 4.5.

Let $R$ be a differentiably admissible $\mathbb{K}$-algebra. Let $f,g\in R$ be
nonzero elements. There exists $b(s)\in\mathbb{K}[s]\setminus\\{0\\}$ and
$\delta[s]\in D_{R|\mathbb{K}}[s]$ such that

$\delta(s)\frac{f}{g}{\bf\left(\frac{f}{g}\right)^{s}}=b(s){\bf\left(\frac{f}{g}\right)^{s}}$

which is a functional equation in $\mathscr{M}_{f/g}^{1}[s]$.

###### Proof.

By Proposition 4.2, we have the functional equation

$\delta(s)f{\bf\frac{f^{s}}{g^{s+1}}}=b(s){\bf\frac{f^{s}}{g^{s+1}}}$

in $\mathscr{M}_{f/g}^{1}[s]$. Let
$\phi:\mathscr{M}_{f/g}^{1}[s]\to\mathscr{M}_{f/g}^{0}[s]$ be the map defined
by

$\frac{p(s)h}{f^{a}g^{b}}{\bf\frac{f^{s}}{g^{s+1}}}\mapsto\frac{p(s)h}{f^{a}g^{b+1}}{\bf\left(\frac{f}{g}\right)^{s}}.$

We note that $\phi$ is an isomorphism of $D_{R|\mathbb{K}}[s]$-modules. Then,
after appliying $\phi$, we get that

$\delta(s)\frac{f}{g}{\bf\left(\frac{f}{g}\right)^{s}}=b(s)\frac{1}{g}{\bf\left(\frac{f}{g}\right)^{s}}$

The result follows after multiplying by $g$. ∎

######  Definition 4.6.

Let $R$ be a differentiably admissible $\mathbb{K}$-algebra and $f,g\in R$ be
nonzero elements. The Bernstein-Sato polynomial $b_{f/g}(s)\in\mathbb{K}[s]$
of the meromorphic function $f/g$ is the monic polynomial of smallest degree
satisfying the functional equation in Theorem 4.5.

######  Remark 4.7.

In this situation we may interpret the Bernstein-Sato polynomial of $f/g$ as
the minimal polynomial of the action of $s$ on

$\displaystyle\frac{D_{R|\mathbb{K}}[s]{\bf\frac{f^{s}}{g^{s+\alpha}}}}{D_{R|\mathbb{K}}[s]\frac{f}{g}{\bf\frac{f^{s}}{g^{s+\alpha}}}}.$

We may also give a $D_{R[t]|\mathbb{K}}$-module structure where the new
variable $t$ acts now as multiplication by $f/g$.

######  Remark 4.8.

Takeuchi [Tak21] has independently introduced a theory of Bernstein-Sato
polynomials which slightly differs from the one in Definition 4.3. He defines
the Bernstein-Sato polynomial $b_{f/g,\alpha}^{\rm mero}(s)$ of order $\alpha$
of the meromorphic function $f/g$ as the monic polynomial of smallest degree
satisfying the functional equation

(4.0.1)
$\delta_{1}[s]\frac{f}{g}{{\bf\frac{f^{s}}{g^{s+\alpha}}}}+\cdots+\delta_{\ell}(s)\frac{f^{\ell}}{g^{\ell}}{{\bf\frac{f^{s}}{g^{s+\alpha}}}}=b(s){{\bf\frac{f^{s}}{g^{s+\alpha}}}}$

for some $\delta_{i}[s]\in D_{R|\mathbb{K}}[s]$ and $\ell\geq 1$ large enough.
He points out that one may also consider the functional equation

(4.0.2)
$\delta(s)\frac{f}{g}{{\bf\frac{f^{s}}{g^{s+\alpha}}}}=b(s){{\bf\frac{f^{s}}{g^{s+\alpha}}}},$

which would lead to a Bernstein-Sato polynomial that we temporaly denote by
$b_{f/g,\alpha}(s)$. They satisfy the following relation

$b_{f/g,\alpha}^{\rm mero}(s)\hskip 5.69054pt|\hskip
5.69054ptb_{f/g,\alpha}(s).$

When comparing to our family of Bernstein-Sato polynomials of order $\alpha$,
which satisfy the functional equation

(4.0.3)
$\delta(s)f{{\bf\frac{f^{s}}{g^{s+\alpha}}}}=b(s){{\bf\frac{f^{s}}{g^{s+\alpha}}}},$

we have

$b_{f/g}^{\alpha}(s)\hskip 5.69054pt|\hskip 5.69054ptb_{f/g,\alpha}(s).$

We point out that Takeuchi’s $b_{f/g,0}(s)$ coincides with our $b_{f/g}(s)$
and, following the proof of Theorem 4.5, we have

$b_{f/g}(s)\hskip 5.69054pt|\hskip 5.69054ptb_{f/g}^{1}(s).$

######  Example 4.9 (Separated variables).

Take two sets of independent variables $x=(x_{1},\ldots,x_{k})$ and
$y=(y_{1},\ldots,y_{l})$, i.e. $x_{i}\neq y_{j}$ for every $i=1,\ldots,k$ and
$j=1,\ldots,l$. Let $b_{f}(s)$ be the Bernstein- Sato polynomial of $f(x)$,
and let $P(s,x,\partial_{x})$ be a differential operator such that

$P(s,x,\partial_{x})f^{s+1}=b_{f}(s)f^{s}.$

Then for any $\alpha\in\mathbb{R}_{\geq 0}$ and $g(y)$ we have

$P(s,x,\partial_{x})f{\bf\frac{f^{s}}{g^{s+\alpha}}}=b_{f}(s){\bf\frac{f^{s}}{g^{s+\alpha}}},$

and thus $b_{f}(s)\mid b_{f/g}^{\alpha}(s)$.

######  Example 4.10.

As a particular case of the preceding example, consider the function

$h(x_{1},\ldots,x_{n})=\frac{x_{1}^{m_{1}}\cdots
x_{k}^{m_{k}}}{x_{k+1}^{m_{k+1}}\cdots x_{n}^{m_{n}}}.$

Then for every $\alpha\in\mathbb{R}_{\geq 0}$ the polynomial
$b^{\alpha}_{h}(s)$ equals

$\prod_{i=1}^{k}\left(\prod_{j=1}^{m_{i}}\left(s+\frac{j}{m_{i}}\right)\right).$

######  Remark 4.11.

Since the operator $\delta(s)=1$ verifies the functional equation

$\delta(s)\frac{1}{g^{s+\alpha}}=1\frac{1}{g^{s+\alpha}},$

we have that $b^{\alpha}_{1/g}(s)=1$ for every $\alpha\in\mathbb{R}_{\geq 0}$,
and thus the Bernstein-Sato polynomial of $1/g$ has no roots. However, there
exist non constant functions $g$ for which the topological zeta function of
$1/g$ has non trivial poles [GVL14]. This phenomenon shows that a direct
generalization of the strong monodromy conjecture in the case of meromorphic
functions is not possible and deserves further investigation.

We now show that $-1$ is always a root of $b^{\alpha}_{f/g}$ in most cases
(see also Remark 4.11).

######  Proposition 4.12.

Let $R$ be a differentiably admissible $\mathbb{K}$-algebra and $f,g\in R$ be
nonzero elements such that $f$ is a nonzero divisor in $R/gR$ nor a unit.
Then, $b^{\alpha}_{f/g}(-1)=0$ for every $\alpha\in\mathbb{R}$.

###### Proof.

We fix $\alpha\in\mathbb{R}$. We set
$M^{\alpha}_{f/g}=R_{fg}{\bf\frac{1}{g^{\alpha}}}$. This free module has an
structure of $D_{R|\mathbb{K}}$-module given by

$\partial\cdot\frac{h}{f^{a}g^{b}}{\bf\frac{1}{g^{\alpha}}}=\partial\left(\frac{h}{f^{a}g^{b}}\right){\bf\frac{1}{g^{\alpha}}}-\alpha\frac{\delta(g)}{f^{a}g^{b+1}}{\bf\frac{1}{g^{\alpha}}}$

where $\partial$ is a $\mathbb{K}$-linear derivative on $R$. There is a
specialization map of $D_{R|\mathbb{K}}$-modules from
$\mathscr{M}^{\alpha}_{f/g}[s]\to M^{\alpha}_{f/g}$ by sending $s\mapsto-1$.

There exists $\delta(s)\in D_{R|\mathbb{K}}[s]$ such that

(4.0.4)
$\delta(s)f{\bf\frac{f^{s}}{g^{s+\alpha}}}=b(s){\bf\frac{f^{s}}{g^{s+\alpha}}}.$

After specializing, we have

(4.0.5)
$\delta(-1){\bf\frac{1}{g^{\alpha}}}=b(-1)\frac{1}{f}{\bf\frac{1}{g^{\alpha}}}.$

We note that $\delta(-1){\bf\frac{1}{g^{\alpha}}}\in
R_{g}{\bf\frac{1}{g^{\alpha}}}$. Then, there exists $u\in R$ and
$\theta\in\mathbb{N}$ such that
$\delta(-1){\bf\frac{1}{g^{\alpha}}}=\frac{u}{g^{\theta}}{\bf\frac{1}{g^{\alpha}}}$.
Then Equation 4.0.5 implies $\frac{u}{g^{\theta}}=b(-1)\frac{1}{f}$, which is
equivalent to $fu=b(-1)g^{\theta}$. If $\theta=0$, we have that $b(-1)=0$,
because $f$ is not a unit. Since $f$ is a nonzero divisor in $R/gR$, it is
also a nonzero divisor in $R/g^{\theta}R$ if $\theta\neq 0.$ Thus
$b^{\alpha}_{f/g}(-1)=0$. ∎

######  Corollary 4.13.

Let $R=\mathbb{C}[x_{1},\ldots,x_{d}]$ and $f,g\in R$ be nonzero elements such
that $f$ does not divide $g$. Then, $b^{\alpha}_{f/g}(-1)=0$ for every
$\alpha\in\mathbb{R}$.

###### Proof.

We can assume that $f$ and $g$ are relatively prime. If $g$ is a unit, this is
a known fact of the classical Bernstein-Sato polynomial. Then, the claim
follows from Proposition 4.12, because $f$ and $g$ form a regular sequence. We
note that $f$ is not a unit because it does not divide $g$. ∎

###  4.1. Rationality of the roots of the meromorphic Bernstein-Sato
polynomial.

A classical result of Kashiwara [Kas77] asserts that the roots of the
Bernstein-Sato polynomial are negative rational numbers. This result was later
refined by Lichtin [Lic89] by giving a set of candidate roots described in
terms of the numerical data of the log resolution of the singularity. A
similar result also holds in the meromorphic case.

Let ${\mathcal{O}}_{\mathbb{C}^{n},0}$ be the ring of germs of holomorphic
functions around a point $0\in\mathbb{C}^{n}$, which we identify with
$R=\mathbb{C}\\{x_{1},\dots,x_{n}\\}$ by taking local coordinates. We take a
small neighborhood of the origin $U\subseteq\mathbb{C}^{n}$ where $f$ and $g$
are holomorphic. Let $J_{f}=(\frac{df}{dx_{1}},\dots,\frac{df}{dx_{1}})$ and
$J_{g}=(\frac{dg}{dx_{1}},\dots,\frac{dg}{dx_{1}})$ be the Jacobian ideals of
$f$ and $g$ respectively. We may assume that the zero locus of $J_{f}$ is
contained in the zero locus of $f$ and the same condition holds for $g$.

Let $\pi:X\to U$ be a log resolution of the meromorphic germ $f/g$, which is
in particular a log resolution of $f^{-1}(0)\cup g^{-1}(0)$. In the sequel we
will use the notations in Section 2.4 but just recall that we associated to
$f/g$ the divisor $\tilde{F}=\tilde{F}_{0}+\tilde{F}_{\infty}$ where

$\tilde{F}_{0}=\sum_{i\in I_{0}}N_{f/g,i}E_{i}\hskip
14.22636pt\textrm{and}\hskip 14.22636pt\tilde{F}_{\infty}=\sum_{i\in
I_{\infty}}N_{f/g,i}E_{i}.$

######  Theorem 4.14.

Let $f,g\in R$ be nonzero holomorphic functions and $\alpha\in\mathbb{R}_{\geq
0}$. The roots of $b_{f/g}^{\alpha}(s)$ are negative rational numbers.
Moreover, they are contained in the set

$\Bigg{\\{}\frac{k_{i}+1+\ell}{N_{f/g,i}}\;;\;\ell\geq 0,\;i\in
I_{0}\Bigg{\\}}.$

###### Proof.

Let $\pi:X\to U$ be a log resolution of the meromorphic germ $f/g$ and let
$D_{X}$ and $D$ denote the corresponding rings of differential operators.
Assume that in local coordinates around a point $p\in X$ we have either

$\frac{\tilde{f}}{\tilde{g}}=\;z_{1}^{N_{1}}\cdots z_{m}^{N_{m}}\hskip
14.22636pt\textrm{or}\hskip
14.22636pt\frac{\tilde{f}}{\tilde{g}}=\;\frac{1}{z_{1}^{N_{1}}\cdots
z_{m}^{N_{m}}}$

Therefore, locally at the point $p$, we have that the Bernstein-Sato
polynomial of order $\alpha$ is either

$b_{\tilde{f}/\tilde{g}}^{\alpha}(s)=\prod_{i=1}^{m}\left(\prod_{j=1}^{N_{i}}\left(s+\frac{j}{N_{i}}\right)\right)\hskip
14.22636pt\textrm{or}\hskip 14.22636ptb_{\tilde{f}/\tilde{g}}^{\alpha}(s)=1.$

By considering a cover of $X$ by affine open subsets where we can use local
coordinates as above, we obtain the global Bernstein-Sato polynomial, that we
also denote by $b_{\tilde{f}/\tilde{g}}^{\alpha}(s)$ if no confusion arises,
as the least common multiple of the local Bernstein-Sato polynomials. Then the
rationality result boils down to proving the existence of an integer $\ell\geq
0$ such that

(4.1.1)
$b_{f/g}^{\alpha}(s)\;|\;b_{\tilde{f}/\tilde{g}}^{\alpha}(s)b_{\tilde{f}/\tilde{g}}^{\alpha}(s+1)\cdots
b_{\tilde{f}/\tilde{g}}^{\alpha}(s+\ell).$

The proof of this fact in the meromorphic case is analogous to the classical
case considered by Kashiwara. We only have to be careful when dealing with the
module

$\mathscr{N}_{\tilde{f}/\tilde{g}}^{\alpha}=D_{X}[s]{\bf\frac{\tilde{f}^{s}}{\tilde{g}^{s+\alpha}}}$

If $\frac{\tilde{f}}{\tilde{g}}=\;z_{1}^{N_{1}}\cdots z_{m}^{N_{m}}$ we have
that $\mathscr{N}_{\tilde{f}/\tilde{g}}^{\alpha}$ is subholonomic as
$D_{X}$-module as in the classical case. Specifically, its characteristic
variety is the closure of $\\{(x,sdf(x))\;;\;f(x)\neq 0,s\in\mathbb{C}\\}$ in
the cotangent bundle $T^{\ast}X$ and thus it has dimension $n+1$.
$\mathscr{N}_{\tilde{f}/\tilde{g}}^{\alpha}$ has the same characteristic
variety when $\frac{\tilde{f}}{\tilde{g}}=\;\frac{1}{z_{1}^{N_{1}}\cdots
z_{m}^{N_{m}}}$ so it is subholonomic as well.

The rest of the proof follows the same lines of reasoning of Kashiwara so we
will just sketch the key points and refer to [Kas77] for details. Let

$\mathscr{N}=\mathbb{R}^{0}\pi_{+}\mathscr{N}_{\tilde{f}/\tilde{g}}^{\alpha}=\mathbb{R}^{0}\pi_{\ast}(D_{U\leftarrow
X}\otimes_{D}^{\bf L}\mathscr{N}_{\tilde{f}/\tilde{g}}^{\alpha})$

be the degree zero direct image of
$\mathscr{N}_{\tilde{f}/\tilde{g}}^{\alpha}$, where $D_{U\leftarrow X}$
denotes the transfer bimodule. There is a canonical section $u\in\mathscr{N}$
associated to ${\bf 1}_{U\leftarrow
X}\otimes{\bf\frac{\tilde{f}^{s}}{\tilde{g}^{s+\alpha}}}\in D_{U\leftarrow
X}\otimes_{D}^{\bf L}\mathscr{N}_{\tilde{f}/\tilde{g}}^{\alpha}$ that allows
us to describe a $D$-submodule

$\mathscr{N}^{\prime}=D[s]u\subseteq\mathscr{N}$

This inclusion is indeed a morphism of $D\langle s,t\rangle$-modules. We have
that $\mathscr{N}$ is subholonomic as $D$-module [Kas77, Lemma 5.7] and
moreover $\mathscr{N}/t\mathscr{N}$ is holonomic so the action of $s$ has a
minimal polynomial which we denote $b_{\mathscr{N}}(s)$. We have that
$\mathscr{N}^{\prime}/t\mathscr{N}^{\prime}$ is holonomic as well which leads
to a polynomial $b_{\mathscr{N}^{\prime}}(s)$. Using that
$\mathscr{N}/\mathscr{N}^{\prime}$ is also holonomic we get by [Kas77,
Proposition 5.11] that there exist $\ell\geq 0$ large enough such that
$t^{\ell}\mathscr{N}\subseteq\mathscr{N}^{\prime}$. From the relation

$b_{\mathscr{N}}(s+j)t^{j}\mathscr{N}=t^{j}b_{\mathscr{N}}(s)\mathscr{N}\subseteq
t^{j+1}\mathscr{N},$

$j\geq 0$, we obtain

$b_{\mathscr{N}}(s+\ell)\cdots b_{\mathscr{N}}(s)\mathscr{N}^{\prime}\subseteq
b_{\mathscr{N}}(s+\ell)\cdots b_{\mathscr{N}}(s)\mathscr{N}\subseteq
t^{\ell+1}\mathscr{N}\subseteq t\mathscr{N}^{\prime}$

and thus $b_{\mathscr{N}^{\prime}}(s)\;|\;b_{\mathscr{N}}(s+\ell)\cdots
b_{\mathscr{N}}(s)$. Finally, we have to relate this equation to (4.1.1). On
the one hand, since
$b_{\tilde{f}/\tilde{g}}^{\alpha}(s)\mathscr{N}_{\tilde{f}/\tilde{g}}^{\alpha}\subseteq
t\mathscr{N}_{\tilde{f}/\tilde{g}}^{\alpha}$ there exist a $D_{X}$-module
endomorphism
$h:\mathscr{N}_{\tilde{f}/\tilde{g}}^{\alpha}\rightarrow\mathscr{N}_{\tilde{f}/\tilde{g}}^{\alpha}$
such that $b_{\tilde{f}/\tilde{g}}^{\alpha}(s)=t\circ h$. Applying the functor
$\mathbb{R}^{0}\pi_{+}$ we get

$b_{\tilde{f}/\tilde{g}}^{\alpha}(s)\mathscr{N}=t\circ\mathbb{R}^{0}\pi_{+}(h)\mathscr{N}\subseteq
t\mathscr{N}$

and thus $b_{\mathscr{N}}(s)\;|\;b_{\tilde{f}/\tilde{g}}^{\alpha}(s)$. On the
other hand we have a surjection of $D\langle s,t\rangle$-modules
$\mathscr{N}^{\prime}\twoheadrightarrow\mathscr{N}_{f/g}^{\alpha}$ and thus
$b_{\mathscr{N}^{\prime}}(s)\mathscr{N}_{f/g}^{\alpha}\subseteq
t\mathscr{N}_{f/g}^{\alpha}$ which gives
$b_{f/g}^{\alpha}(s))\;|\;b_{\mathscr{N}^{\prime}}(s)$ and Equation (4.1.1)
follows.

The refinement given by Lichtin [Lic89] requires to pass from left to right
$D$-modules in order to plug in all the information given by the relative
canonical divisor. Namely, for a left $D$-module $\mathscr{M}$ we consider the
right $D$-module $\mathscr{M}^{(r)}:=\omega_{R}\otimes_{R}\mathscr{M}$. In
local coordinates, this operation is given by the involution on $D$ that sends
a differential operator $\delta$ to its adjoint $\delta^{\ast}$ described
uniquely by the properties
$(\delta\delta^{\prime})^{\ast}=\delta^{\prime\ast}\delta^{\ast}$,
$f^{\ast}=f$ for any $f\in R$ and $\partial_{i}^{\ast}=-\partial_{i}$. This
gives an equivalence between left and right $D$-modules. We can extend this to
an equivalence between left and right $D\langle s,t\rangle$-modules by taking
$t^{\ast}=t$, $\partial_{t}^{\ast}=-\partial_{t}$ and
$s^{\ast}=(-\partial_{t}t)^{\ast}=t\partial_{t}=\partial_{t}t-1=-s-1$. For an
element $u$ in a $D\langle s,t\rangle$-module $\mathscr{M}$ we set
$u^{\ast}=dx_{1}\wedge\cdots\wedge d{x_{n}}\otimes u$. Then we have
$u^{\ast}\delta^{\ast}=(\delta u)^{\ast}$ for any $\delta\in D\langle
s,t\rangle$.

Now, for $u={\bf\frac{f^{s}}{g^{s+\alpha}}}\in\mathscr{N}_{f/g}^{\alpha}$ we
have that the functional equation

$\delta(s)f{\bf\frac{f^{s}}{g^{s+\alpha}}}=b(s){\bf\frac{f^{s}}{g^{s+\alpha}}}\hskip
14.22636pt\textrm{becomes}\hskip
14.22636ptf\left({\bf\frac{f^{s}}{g^{s+\alpha}}}\right)^{\ast}(\delta(s))^{\ast}=(b(s))^{\ast}\left({\bf\frac{f^{s}}{g^{s+\alpha}}}\right)^{\ast}$

and the minimal polynomials satisfying these equations are related by

(4.1.2)
$b_{f/g}^{\alpha}(s)=b_{u}(s)=b_{u^{\ast}}(-s-1)=b_{(f/g)^{\ast}}^{\alpha}(-s-1).$

In particular $b_{u^{\ast}}(s)$ is the minimal polynomial of the action of $s$
on $(\mathscr{N}_{f/g}^{\alpha})^{(r)}/(\mathscr{N}_{f/g}^{\alpha})^{(r)}t$.

Let $\pi:X\to U$ be a log resolution of $f/g$. We may construct
$(\mathscr{N}_{\tilde{f}/\tilde{g}}^{\alpha})^{(r)}=\omega_{X}\otimes_{{\mathcal{O}}_{X}}\mathscr{N}_{\tilde{f}/\tilde{g}}^{\alpha}$
as before but, following Lichtin, we consider the $D_{X}\langle
s,t\rangle$-submodule

$\mathscr{N}_{v}:=vD_{X}[s]$

with $v=\pi^{\ast}(dx_{1}\wedge\cdots\wedge
dx_{n})\otimes{\bf\frac{\tilde{f}^{s}}{\tilde{g}^{s+\alpha}}}\in\left(\mathscr{N}_{\tilde{f}/\tilde{g}}^{\alpha}\right)^{(r)}$.
In local coordinates around a point $p\in X$ we have either

$v=\;wz_{1}^{N_{1}s+k_{1}}\cdots z_{m}^{N_{m}s+k_{m}}\hskip
14.22636pt\textrm{or}\hskip 14.22636ptv=\;wz_{1}^{-N_{1}s+k_{1}}\cdots
z_{m}^{-N_{m}s+k_{m}},$

where $w$ is a unit. The Bernstein-Sato polynomial associated to these
elements is either

$b_{v}(s)=\prod_{i=1}^{m}\left(\prod_{j=1}^{N_{i}}\left(s+\frac{k_{i}+j}{N_{i}}\right)\right)\hskip
14.22636pt\textrm{or}\hskip 14.22636ptb_{v}(s)=1,$

and we have a relation with the Bernstein-Sato polynomial associated to
$\mathscr{N}_{v}$, i.e. the minimal polynomial of the action of $s$ on
$\mathscr{N}_{v}/\mathscr{N}_{v}t$, given by
$b_{\mathscr{N}_{v}}(s)=b_{v}(-s-1)$. By considering a cover of $X$ we get the
global Bernstein-Sato polynomial that we also denote by
$b_{\mathscr{N}_{v}}(s)$ if no confusion arises.

Then the refined version of Lichtin result follows from proving the existence
of an integer $\ell\geq 0$ such that the polynomial $b_{u^{\ast}}(s)$ in
Equation (4.1.2) satisfies

(4.1.3)
$b_{u^{\ast}}(s)\;|\;b_{\mathscr{N}_{v}}(s)b_{\mathscr{N}_{v}}(s-1)\cdots
b_{\mathscr{N}_{v}}(s-\ell).$

The rest of the proof is analogous to Kashiwara’s proof of rationality but
working in the category of right $D$-modules.

∎

######  Corollary 4.15.

Let $f,g\in R$ be nonzero holomorphic functions. The roots of $b_{f/g}(s)$ are
negative rational numbers. Moreover, they are contained in the set

$\Bigg{\\{}\frac{k_{i}+1+\ell}{N_{f/g,i}}\;;\;\ell\geq 0,\;i\in
I_{0}\Bigg{\\}}.$

###### Proof.

It follows from the fact that $b_{f/g}(s)$ divides $b_{f/g}^{1}(s)$. ∎

##  5\. Archimedean Local Zeta Functions

In this section we use the Bernstein-Sato polynomial of $f/g$ to study
Archimedean local zeta functions. Let ${\mathcal{O}}_{\mathbb{C}^{n},0}$ be
the ring of germs of holomorphic functions around a point
$0\in\mathbb{C}^{n}$, which we identify with
$R=\mathbb{C}\\{x_{1},\dots,x_{n}\\}$ by taking local coordinates. For
definiteness we take a small neighborhood of the origin
$U\subseteq\mathbb{C}^{n}$ where $f$ and $g$ are both holomorphic. By a test
function we mean a smooth function with compact support on $\mathbb{C}^{n}$.
Then the local zeta function attached to a test function $\phi$ and $f/g$ is
defined as

$Z_{\phi}(s,f/g)=\bigintssss_{U\setminus
H}\phi(x)\left|\frac{f(x)}{g(x)}\right|^{2s}\ \mathrm{d}x,$

where $s$ is a complex number and $H=f^{-1}(0)\cup g^{-1}(0)$. When $g=1$ (the
classical case), local zeta functions were introduced by Gel’fand and Shilov
in the 50’s [GS16]. In that case is not difficult to show that
$Z_{\phi}(s,f/1)=Z_{\phi}(s,f)$ converges on the half plane
$\\{s\in\mathbb{C}\ ;\operatorname{Re}(s)>0\\}$ and defines a holomorphic
function there. In contrast, the convergence of the parametric integral
$Z_{\phi}(s,f/g)$ is a delicate matter as Veys and Zúñiga-Galindo noted
[VZG17, Remark 2.1 (1)]. In particular they explain that the convergence of
the integral does not follow from the fact that $\phi$ has compact support and
they use an embedded resolution of $H$ to show that the integral has a region
of convergence [VZG17, Theorem 3.5 (1)]. As in the classical case, the
meromorphic continuation of $Z_{\phi}(s,f/g)$ does not depend on the set $U$.

Building on the properties of the log canonical threshold of $f$ and $g$ we
describe below a simple region of convergence for $Z_{\phi}(s,f/g)$. First, we
recall the definition of the log canonical threshold of $f$ and some of its
properties regarding the local zeta function of $f$ and $\phi$.

######  Definition 5.1.

Let $f$ be a holomorphic function in an open set $U\subseteq\mathbb{C}^{n}$.
The log-canonical threshold of $f$ (at the origin) is the number

$\operatorname{lct}_{0}(f)=\sup\left\\{\lambda\in\mathbb{R}_{>0}\ ;\
\int_{B_{\varepsilon}(0)}|f|^{-2\lambda}<\infty,\text{ for some
}\varepsilon>0\right\\}.$

It is known that $\operatorname{lct}_{0}(f)$ is a positive rational number.
Furthermore, we have that $Z_{\phi}(s,f)$ is an holomorphic function on
$\operatorname{Re}(s)>-\operatorname{lct}_{0}(f)$. Using this fact one may
show that $Z_{\phi}(s,g^{-1})=\int_{U\setminus
g^{-1}(0)}\phi(x)\left|g(x)\right|^{-2s}\ \mathrm{d}x$, is holomorphic on
$\operatorname{Re}(s)<\operatorname{lct}_{0}(g)$.

######  Lemma 5.2.

Let $f$ and $g$ be nonzero holomorphic functions. Then, the integral
$Z_{\phi}(s,f/g)$ converges for
$-\operatorname{lct}_{0}(f)<\operatorname{Re}(s)<\operatorname{lct}_{0}(g)$.
Furthermore, it defines a holomorphic function there.

###### Proof.

First we show that $Z_{\phi}(s,f/g)$ is finite on the interval
$-\operatorname{lct}_{0}(f)<\operatorname{Re}(s)\leq 0$. Note that in this
region the function $|g(x)|^{-2s}$ is well defined and continuous over
$U\setminus H$. Then

$\left|\bigintssss_{U\setminus H}\phi(x)\left|\frac{f(x)}{g(x)}\right|^{2s}\
\mathrm{d}x\right|\leq\bigintssss_{U\setminus
H}|\phi(x)|\left|\frac{f(x)}{g(x)}\right|^{2\operatorname{Re}(s)}\
\mathrm{d}x<\infty,$

since the the support of $\phi$ is compact and
$\left|\frac{f(x)}{g(x)}\right|^{2\operatorname{Re}(s)}$ is continuous. A
symmetric argument shows that $Z_{\phi}(s,f/g)<\infty$ for
$0\leq\operatorname{Re}(s)<\operatorname{lct}_{0}(g)$. The fact that
$Z_{\phi}(s,f/g)$ defines a holomorphic function on
$-\operatorname{lct}_{0}(f)<\operatorname{Re}(s)<\operatorname{lct}_{0}(g)$
can be proved as in the classical case by showing that
$\partial/\partial\bar{s}(Z_{\phi}(s,f/g))=0$, which follows in particular
from Lebesgue’s dominated convergence theorem [Igu78, Theorem 3.1]. ∎

######  Remark 5.3.

The region of convergence of Lemma 5.2 is in general smaller that the region
of convergence described in the work of Veys and Zuñiga-Galindo [VZG17].
According to their Example 3.13 (4), the integral
$Z_{\phi}(s,y^{2}+x^{4}/x^{2}+y^{4})$ converges in
$-3/2<\operatorname{Re}(s)<3/2$, whereas Lemma 5.2 shows that the integral
converges in $-3/4<\operatorname{Re}(s)<3/4$. We point out that the region of
convergence of Veys and Zuñiga-Galindo is optimal in the sense that the
endpoints give actual poles for $Z_{\phi}(s,f/g)$, i.e. $-3/2$ and $3/2$ are
poles of $Z_{\phi}(s,y^{2}+x^{4}/x^{2}+y^{4})$.

Another property of $Z_{\phi}(s,f)$ that one may also study is the existence
of a meromorphic continuation to the whole complex plane. This fact was
conjectured by Gel’fand in the 50’s and proved by Bernstein and Gel’fand
[BG69], Atiyah [Ati70] using resolution of singularities, and Bernstein
[Ber72] using the Bernstein-Sato polynomial. In the case of meromorphic
functions, Veys and Zúñiga-Galindo [VZG17, Theorem 3.5 (2)] showed that
$Z_{\phi}(s,f/g)$ has a meromorphic continuation to the whole complex plane
and its poles are described by means of the numerical data of an embedded
resolution of singularities of $H$. By using the theory of Bernstein-Sato
polynomials developed in Sections 3 and 4 we present a proof of the
meromorphic continuation of $Z_{\phi}(s,f/g)$, resembling the original proof
of Bernstein [Ber72], giving in addition a shorter list of candidate poles.

For a differential operator $\delta(s)\in D_{R|\mathbb{C}}[s]$ we denote the
conjugate operator by $\overline{\delta}(s)$. This is the operator obtained
from $\delta$ by replacing $x_{i}$ with $\overline{x_{i}}$ and
$\partial_{x_{i}}$ with $\partial_{\overline{x_{i}}}$. A simple calculation
shows that the functional equation in Lemma 4.2 implies

(5.0.1)
$\delta(s)\overline{\delta}(s)\frac{|f|^{2(s+1)}}{|g|^{2(s+\alpha)}}=b^{\alpha}_{f/g}(s)^{2}\frac{|f|^{2s}}{|g|^{2(s+\alpha)}}\text{
in }\mathscr{M}_{f/g}^{\alpha}[s].$

######  Theorem 5.4.

Let $f$ and $g$ be nonzero holomorphic functions and take
$\alpha=\operatorname{lct}_{0}g$. The local zeta function $Z_{\phi}(s,f/g)$
has a meromorphic continuation to the whole complex plane $\mathbb{C}$, and
its poles are contained in the set

$\Bigl{\\{}\zeta-k\alpha\ ;\ k\in\mathbb{Z}_{\geq 0}\Bigr{\\}}_{\zeta\text{
root of }b^{\alpha}_{f/g}(s)}\bigcup\Bigl{\\{}k\alpha-\xi\ ;\
k\in\mathbb{Z}_{\geq 0}\Bigr{\\}}_{\xi\text{ root of }b^{\alpha}_{g/f}(s)}.$

In particular, the poles of $Z_{\phi}(s,f/g)$ are rational numbers.

###### Proof.

Let $b^{\alpha}_{f/g}(s)\in\mathbb{C}[s]$ be the nonzero polynomial of Lemma
4.2, note that there exists a map of $D_{R|\mathbb{C}}$-modules

$\psi_{\lambda}:\mathscr{M}_{f/g}^{\alpha}[s]\to\operatorname{Frac}(\mathbb{C}\\{x_{1},\ldots,x_{d}\\})$

that maps $s\mapsto\lambda$ and
${\bf\frac{f^{s}}{g^{s+\alpha}}}\mapsto\frac{f^{\lambda}}{g^{\lambda+\alpha}}$
for every $\lambda\in\mathbb{C}$. We also notice that after possible shrinking
$U$, (5.0.1) remains valid for $f,g\in R$, and then multiplication of
$Z_{\phi}(s,f/g)$ by $b^{\alpha}_{f/g}(s)^{2}$ gives

$\displaystyle b^{\alpha}_{f/g}(s)^{2}\,Z_{\phi}(s,f/g)$
$\displaystyle=\bigintssss_{U\setminus
H}\phi(x)b^{\alpha}_{f/g}(s)^{2}\,\left|\frac{f(x)}{g(x)}\right|^{2s}\,\mathrm{d}x$
$\displaystyle=\bigintssss_{U\setminus
H}\phi(x)|g(x)|^{2\alpha}\,b^{\alpha}_{f/g}(s)^{2}\frac{1}{|g(x)|^{2\alpha}}\left|\frac{f(x)}{g(x)}\right|^{2s}\,\mathrm{d}x$
$\displaystyle=\bigintssss_{U\setminus
H}\phi(x)|g(x)|^{2\alpha}\,\left(\delta(s)\overline{\delta}(s)\frac{|f(x)|^{2(s+1)}}{|g(x)|^{2(s+\alpha)}}\right)\,\mathrm{d}x$
$\displaystyle=\bigintssss_{U\setminus
H}\phi(x)|g(x)|^{2\alpha}\,\left(\delta(s)\overline{\delta}(s)|f(x)|^{2-2\alpha}\left|\frac{f(x)}{g(x)}\right|^{2(s+\alpha)}\right)\,\mathrm{d}x$
$\displaystyle=\bigintssss_{U\setminus
H}\left(|f(x)|^{2-2\alpha}\delta^{\ast}(s)\overline{\delta}^{\ast}(s)\cdot\phi(x)|g(x)|^{2\alpha}\right)\,\left|\frac{f(x)}{g(x)}\right|^{2(s+\alpha)}\,\mathrm{d}x$
$\displaystyle=Z_{\psi}(s+\alpha,f/g).$

Here $\delta^{\ast}$ denotes the adjoint operator of $\delta$ and
$\psi=|f(x)|^{2-2\alpha}\delta^{\ast}(s)\overline{\delta}^{\ast}(s)\cdot\phi(x)|g(x)|^{2\alpha}$
is a complex test function. By Lemma 5.2, $Z_{\psi}(s+\alpha,f/g)$ converges
for $\beta-\alpha<\operatorname{Re}(s)<0$ and thus

$Z_{\phi}(s,f/g)=\frac{Z_{\psi}(s+\alpha,f/g)}{b^{\alpha}_{f/g}(s)^{2}}$

converges in $\beta-\alpha<\operatorname{Re}(s)<\alpha$ outside the possible
roots of $b_{f/g}(s)$. If we now multiply $Z_{\psi}(s+\alpha,f/g)$ by
$b^{\alpha}_{f/g}(s+\alpha)^{2}$ and repeat the reasoning above, we may
conclude that $Z_{\phi}(s,f/g)$ can be extended meromorphically to
$\beta-2\alpha<\operatorname{Re}(s)<\alpha$ with possible poles in the roots
of $b^{\alpha}_{f/g}(s)\cdot b^{\alpha}_{f/g}(s+\alpha)$. Iterating this
process, we obtain a meromorphic extension to $\operatorname{Re}(s)<\alpha$
for $Z_{\phi}(s,f/g)$, and the possible poles of the zeta function are
contained in the set

$\Bigl{\\{}\zeta-k\alpha\ ;\ k\in\mathbb{Z}_{\geq 0}\Bigr{\\}}_{\zeta\text{
root of }b^{\alpha}_{f/g}(s)}.$

For the continuation ‘to the right’ of $Z_{\phi}(s,f/g)$ we use Lemma 4.2 in
the following version

$\gamma(-s)\frac{f^{\alpha-s}}{g^{1-s}}=b^{\alpha}_{g/f}(-s)\frac{f^{s-\alpha}}{g^{s}},$

which implies the analogue of 5.0.1:
$\gamma(-s)\overline{\gamma}(-s)\frac{|f|^{2(\alpha-s)}}{|g|^{2(1-s)}}=b^{\alpha}_{g/f}(-s)^{2}\frac{|f|^{2(s-\alpha)}}{|g|^{2s}}$.
If we multiply $Z_{\phi}(s,f/g)$ by $b^{\alpha}_{g/f}(-s)^{2}$ we obtain

$Z_{\phi}(s,f/g)=\frac{Z_{\varphi}(\alpha-s,f/g)}{b^{\alpha}_{g/f}(-s)^{2}},$

showing that $Z_{\phi}(s,f/g)$ converges in
$\beta<\operatorname{Re}(s)<\alpha-\beta$ away from the possible roots of
$b^{\alpha}_{g/f}(s)$. Further iteration of this process provides the desired
conclusion. The rationality of the poles of $Z_{\phi}(s,f/g)$ follows from
Theorem 4.14. ∎

######  Remark 5.5.

Veys and Zúñiga-Galindo study local zeta functions for meromorphic functions
over local fields of characteristic zero [VZG17]. The statement and proof of
Lemma 5.2 are also valid in this generality, providing also a simple region of
convergence for the non Archimedean $Z_{\phi}(s,f/g)$, cf. [VZG17, Theorem 3.2
(1)]. We do not known if the theory developed in sections 3 and 4 is also
valid in this generality, but it is certainly true over Archimedean local
fields of characteristic zero, that is, $\mathbb{R}$ or $\mathbb{C}$. The
proof of Theorem 5.4 for the real case can be given by adapting the ideas of
our proof and following the lines of Igusa’s work [Igu00, Theorem 5.3.1].

##  6\. Multiplier ideals: analytic construction

In this section we describe the theory of analytic multiplier ideals for
meromorphic functions and relate their jumping numbers to roots of the
meromorphic Bernstein-Sato polynomial. To this end, we use the previous
notation, identifying ${\mathcal{O}}_{\mathbb{C}^{n},0}$ with the ring
$R=\mathbb{C}\\{x_{1},\dots,x_{n}\\}$ by taking local coordinates.

######  Definition 6.1.

Let $f,g\in R$ be nonzero elements such that $f/g$ is not constant and
$\lambda\in\mathbb{R}_{\geq 0}$. We define the multiplier ideal (at the
origin) of $f/g$ at $\lambda$ by

${\mathcal{J}}\left(\left(\frac{f}{g}\right)^{\lambda}\right)=\left\\{h\in R\
;\
\int_{B_{\varepsilon}(0)}\frac{|h|^{2}|g|^{2\lambda}}{|f|^{2\lambda}}<\infty\hbox{
for some }\varepsilon>0\right\\},$

where ${B_{\varepsilon}(0)}$ denotes a closed ball of radius $\varepsilon>0$
around the origin $0$.

The following useful remark shows that we can describe the meromorphic
multiplier ideal from the theory of mixed multiplier ideals
${\mathcal{J}}(f^{\lambda_{1}}g^{\lambda_{2}})$ associated to the pair of
functions.

######  Remark 6.2.

Let $f,g\in R$ be nonzero elements and $\lambda\in\mathbb{R}$. Pick
$t\in\mathbb{N}$ such that $t\geq\lambda$. Then,

$\displaystyle{\mathcal{J}}\left(f^{\lambda}g^{t-\lambda}\right):g^{t}$
$\displaystyle=\left\\{h\in R\ ;\
hg^{t}\in{\mathcal{J}}\left(f^{\lambda}g^{t-\lambda}\right)\right\\}$
$\displaystyle=\left\\{h\in R\ ;\
\int_{B_{\varepsilon}(0)}\frac{|h|^{2}|g|^{2t}}{|f|^{2\lambda}|g|^{2t-2\lambda}}<\infty\hbox{
for some }\varepsilon>0\right\\}$ $\displaystyle=\left\\{h\in R\ ;\
\int_{B_{\varepsilon}(0)}\frac{|h|^{2}|g|^{2\lambda}}{|f|^{2\lambda}}<\infty\hbox{
for some }\varepsilon>0\right\\}$
$\displaystyle={\mathcal{J}}\left(\left(\frac{f}{g}\right)^{\lambda}\right).$

######  Remark 6.3.

Let $f,g\in R$ be nonzero elements and $\lambda\in\mathbb{R}$. Then,

${\mathcal{J}}\left(\left(\frac{f}{g}\right)^{\lambda}\right)={\mathcal{J}}\left(\left(\frac{g}{f}\right)^{-\lambda}\right).$

######  Definition 6.4.

Let $f,g\in R$ be nonzero elements and $\lambda\in\mathbb{R}$. We say that
$\lambda$ is a jumping number for $f/g$ if for every $\varepsilon>0$ we have
either

${\mathcal{J}}\left(\left(\frac{f}{g}\right)^{\lambda}\right)\neq{\mathcal{J}}\left(\left(\frac{f}{g}\right)^{\lambda-\varepsilon}\right)$

or

${\mathcal{J}}\left(\left(\frac{f}{g}\right)^{\lambda}\right)\neq{\mathcal{J}}\left(\left(\frac{f}{g}\right)^{\lambda+\varepsilon}\right).$

We stress that multiplier ideals of meromorphic functions are not necessarily
continuous on neither side. For instance, if $g=1$, it is known that

${\mathcal{J}}\left(f^{\lambda}\right)={\mathcal{J}}\left(f^{\lambda+\varepsilon}\right)$

for small enough $\varepsilon>0$. In contrast, if $f=1$,

${\mathcal{J}}\left(\frac{1}{g^{\lambda}}\right)\neq{\mathcal{J}}\left(\frac{1}{g^{\lambda-\varepsilon}}\right)$

for small enough $\varepsilon>0$.

######  Proposition 6.5.

Let $f,g\in R$ be a nonzero elements and $\lambda\in\mathbb{R}_{\geq 0}$.
Then,

${\mathcal{J}}\left(\left(\frac{f}{g}\right)^{\lambda+\ell}\right)=\left(\frac{f^{\ell}}{g^{\ell}}\left({\mathcal{J}}\left(\frac{f}{g}\right)^{\lambda}\right)\right)\bigcap
R$

for every $\ell\in\mathbb{N}.$ In particular, if $\lambda+1$ is a jumping
number, then $\lambda$ is a jumping number.

###### Proof.

We fix $t\geq\lambda+1$. The result follows from the equality

${\mathcal{J}}\left(f^{\lambda+\ell}g^{t-\lambda-1}\right):g^{t}=\left(\left(\frac{f^{\ell}}{g^{\ell}}{\mathcal{J}}(f^{\lambda}g^{t-\lambda})\right):g^{t}\right)\bigcap
R=\left(\frac{f^{\ell}}{g^{\ell}}\left({\mathcal{J}}(f^{\lambda}g^{t-\lambda}):g^{t}\right)\right)\bigcap
R.$

given by Remark 6.2. ∎

We note that Proposition 6.5 gives a weaker version of Skoda’s Theorem [Laz04,
9.6.21].

######  Theorem 6.6.

Let $f,g\in R$ be nonzero elements and $\lambda\in\mathbb{R}$. Then, the set
of jumping numbers of $f/g$ is a set of rational numbers with no accumulation
points.

###### Proof.

It suffices to show that the set of jumping numbers of $f/g$ in $[0,t]$ is a
finite set of rational numbers for any $t\in\mathbb{Z}_{\geq 0}$. By Remark
6.2, the jumping numbers of ${\mathcal{J}}((\frac{f}{g})^{\lambda})$ is a
subset of the jumping numbers of
${\mathcal{J}}\left(f^{\lambda}g^{t-\lambda}\right)$. The last set is the
intersection of the jumping walls of
${\mathcal{J}}(f^{\lambda_{1}}g^{\lambda_{2}})$ with the line
$\lambda_{1}+\lambda_{2}=t$. We conclude that the set of jumping numbers of
$f/g$ is a set of rational numbers with no accumulation points, because the
jumping walls are the boundaries of the constancy regions of
${\mathcal{J}}(f^{\lambda_{1}}g^{\lambda_{2}})$ and, by construction, they are
defined by linear equations with rational coefficients and there are only
finitely many in any bounded part of the positive orthant
$\mathbb{R}^{2}_{\geq 0}$. ∎

######  Theorem 6.7.

Let $f,g\in R$ be nonzero elements and $\lambda$ be a jumping number of $f/g$
such that $\lambda\in(1-\operatorname{lct}_{0}(g),1]$. Then, $-\lambda$ is a
root of the Bernstein-Sato polynomial of $f/g$.

###### Proof.

Suppose $\lambda>1-\operatorname{lct}_{0}(g)$ is a jumping number of $f/g$ and
take $1-\operatorname{lct}_{0}(g)<\lambda^{\prime}<\lambda$. In particular,
for any $c\in[\lambda^{\prime},\lambda)$ there exist $h$ and $0<r^{\prime}<r$
such that

$\int_{B_{r}(0)}\frac{|h|^{2}|g|^{2c}}{|f|^{2c}}<\infty\quad{\rm
and}\quad\int_{B_{r^{\prime}}(0)}\frac{|h|^{2}|g|^{2\lambda}}{|f|^{2\lambda}}=\infty.$

Similarly to Equation (5.0.1), we deduce that

(6.0.1)
$\delta(s)\overline{\delta}(s)\frac{|f|^{2(s+1)}}{|g|^{2(s+1)}}=b_{f/g}(s)^{2}\frac{|f|^{2s}}{|g|^{2s}}.$

from Theorem 4.5. For $s=-c$ and for any test function $\phi$ supported on
$B_{r}(0)$ we get

(6.0.2) $\displaystyle\int_{B_{r}(0)}|h|^{2}\phi
b_{f/g}(-c)^{2}\frac{|g|^{2c}}{|f|^{2c}}$
$\displaystyle=\int_{B_{r}(0)}|h|^{2}\phi\delta(-c)\overline{\delta}(-c)\frac{|f|^{2(-c+1)}}{|g|^{2(-c+1)}}$
(6.0.3)
$\displaystyle=\int_{B_{\varepsilon}(0)}\frac{|f|^{2(-c+1)}}{|g|^{2(-c+1)}}\delta^{*}(-c)\overline{\delta}^{*}(-c)(|h|^{2}\phi),$

where $\delta^{*}(s)$ and $\overline{\delta}^{*}(s)$ denote the respective
adjoint operators. Since $\lambda^{\prime}>1-\operatorname{lct}_{0}(g)$, the
proof of Lemma 5.2 implies that the right-hand side of Equation (6.0.3) is
uniformly bounded by a positive number depending only on $h$ and $\varphi$.
Assuming that $\phi$ is a test function with support in $B_{r}(0)$ and
identically 1 in the ball $B_{r^{\prime}}(0)$,

$b_{f/g}(-c)^{2}\int_{B_{r^{\prime}}(0)}\frac{|h|^{2}|g|^{2c}}{|f|^{2c}}=\int_{B_{r^{\prime}}(0)}\frac{|f|^{2(-c+1)}}{|g|^{2(-c+1)}}\delta^{*}(-c)\overline{\delta}^{*}(-c)(|h|^{2}).$

We deduce $b_{f/g}(-\lambda)=0$, because
$\int_{B_{r^{\prime}}(0)}\frac{|h|^{2}|g|^{2c}}{|f|^{2c}}$ tends to infinity
as $c$ tends to $\lambda$. ∎

##  7\. Multiplier ideals: algebraic construction

In this section we define the algebraic version of multiplier ideals for
meromorphic functions using log resolutions. Let
${\mathcal{O}}_{\mathbb{C}^{n},0}$ be the ring of germs of holomorphic
functions around a point $0\in\mathbb{C}^{n}$, which we identify with
$R=\mathbb{C}\\{x_{1},\dots,x_{n}\\}$ by taking local coordinates. Throughout
this section we use the notation introduced in Section 2.4 concerning the
numerical data associated to a log resolution $\pi:X\to U$ of the meromorphic
germ $f/g$.

######  Definition 7.1.

Let $f,g\in R$ be nonzero elements such that $f/g$ is not constant and
$\lambda\in\mathbb{R}_{\geq 0}$. We define the $0$-multiplier ideal of $f/g$
at $\lambda$ as the stalk at the origin of

${\mathcal{J}}\left(\left(\frac{f}{g}\right)^{\lambda}\right)=\pi_{*}\mathcal{O}_{X}(-[\lambda\cdot\tilde{F}]+K_{\pi}),$

where $[\cdot]$ denotes the integer part of a real number or
$\mathbb{R}$-divisor. If no confusion arises we denote the stalk at the origin
in the same way and thus
${\mathcal{J}}\left(\left(\frac{f}{g}\right)^{\lambda}\right)\subseteq R$.

######  Remark 7.2.

  1. (1)

The above definition is independent of the choice of the log resolution $\pi$.
This follows from analogous considerations to those in the classical case when
$g=1$ [Laz04, Example 9.1.16, Theorem 9.2.18 & Lemma 9.2.19].

  2. (2)

Given $h\in R$, the condition $h\in{\mathcal{J}}((\frac{f}{g})^{\lambda})$ is
equivalent to

${\rm ord}_{E_{i}}\pi^{*}h\geq[\lambda\cdot N_{f/g,i}]-k_{i}\quad\hbox{ for
every }\,i\in I.$

  3. (3)

Since ${\rm ord}_{E_{i}}\pi^{*}hg^{t}={\rm ord}_{E_{i}}\pi^{*}h+tN_{g,i}$, and
$[\lambda\cdot N_{f/g,i}]+tN_{g,i}=[\lambda\cdot N_{f,i}+(t-\lambda)\cdot
N_{g,i}]$ for all $t\in\mathbb{N}$ such that $t\geq\lambda$, we have that the
condition $h\in{\mathcal{J}}((\frac{f}{g})^{\lambda})$ is equivalent to the
conditions ${\rm ord}_{E_{i}}\pi^{*}hg^{t}\geq[\lambda\cdot
N_{f,i}+(t-\lambda)\cdot N_{g,i}]+k_{i}$ for all $t\in\mathbb{N}$ with
$t\geq\lambda$ and all $i\in I$. This gives a different proof of Remark 6.2.

  4. (4)

Given $h\in R$, we have ${\rm ord}_{E_{i}}\pi^{*}h\geq 0$ for all $i\in I$.
Since ${\mathcal{J}}((\frac{f}{g})^{\lambda})\subseteq R$, it follows that

${\mathcal{J}}\left(\left(\frac{f}{g}\right)^{\lambda}\right)=\pi_{*}\mathcal{O}_{X}(-[\lambda\cdot\tilde{F}_{0}]+K_{\pi}).$

  5. (5)

One can associate $\infty$-multiplier ideals
${\mathcal{J}}^{\infty}((\frac{f}{g})^{\lambda})$ to the meromorphic germ
$f/g$ and the parameter $\lambda$ as follows
${\mathcal{J}}^{\infty}((\frac{f}{g})^{\lambda})={\mathcal{J}}((\frac{g}{f})^{\lambda})$.

The following properties of multiplier ideals
${\mathcal{J}}((\frac{f}{g})^{\lambda})$ are analogous to the case of
multiplier ideals associated to holomorphic germs $f$, that is, when $g=1$.

######  Lemma 7.3.

There exists a discrete strictly increasing sequence of rational numbers

$\lambda_{1}<\lambda_{2}<\cdots$

such that

${\mathcal{J}}\left(\left(\frac{f}{g}\right)^{\lambda_{i+1}}\right)\subsetneq{\mathcal{J}}\left(\left(\frac{f}{g}\right)^{c}\right)={\mathcal{J}}\left(\left(\frac{f}{g}\right)^{\lambda_{i}}\right)$

for $c\in[\lambda_{i},\lambda_{i+1})$, and all $i$. These rational numbers are
called the $0$-jumping numbers of $f/g$.

######  Remark 7.4.

  * •

The candidate $0$-jumping numbers of $f/g$ have the form
$\frac{k_{i}+1+\ell}{N_{f/g,i}}$ with $\ell\in\mathbb{Z}_{\geq 0}$.

  * •

Let $\lambda\in\mathbb{Q}$ be a candidate jumping number of $f$ associated to
the divisor $E_{i}$. Then $\lambda\cdot\frac{N_{f,i}}{N_{f/g,i}}$ is a
candidate $0$-jumping number of $f/g$. Notice that
$\frac{N_{f,i}}{N_{f/g,i}}\geq 1$.

######  Lemma 7.5.

For any $\lambda\in\mathbb{R}_{>0}$ we have
${\mathcal{J}}(f^{\lambda})\subseteq{\mathcal{J}}((\frac{f}{g})^{\lambda})$.
In addition, we have ${\mathcal{J}}((\frac{f}{g})^{n})=(f^{n})$ for any
$n\in\mathbb{Z}_{>0}$.

###### Proof.

The inclusion
${\mathcal{J}}(f^{\lambda})\subseteq{\mathcal{J}}((\frac{f}{g})^{\lambda})$
follows from the fact that $N_{f,i}/N_{f/g,i}\geq 1$. Therefore, we have that
$(f^{n})={\mathcal{J}}(f^{n})\subseteq{\mathcal{J}}((\frac{f}{g})^{n})$.
Moreover, for any $E_{i}$ in the strict transform of $f$ we have that
$N_{f,i}/N_{f/g,i}=1$, and $k_{i}=0$. Hence, if
$h\in{\mathcal{J}}((\frac{f}{g})^{n})$, we have that
$\operatorname{ord}_{E_{i}}\pi^{*}h\geq n\cdot N_{f,i}$. It follows that $f$
divides $h$ and hence ${\mathcal{J}}((\frac{f}{g})^{n})\subseteq(f^{n})$. ∎

######  Remark 7.6.

In general
${\mathcal{J}}((\frac{f}{g})^{\lambda+n})\neq(f^{n})\cdot{\mathcal{J}}((\frac{f}{g})^{\lambda})$,
as Example 7.7 shows. As a consequence, the periodicity of jumping numbers
fails. Alternatively, Skoda’s Theorem for multiplier ideals of meromorphic
germs is weaker than the classic one.

######  Example 7.7.

Let us consider $f=y^{3}+x^{5}$, $g_{1}=x$, $g_{2}=y$, and $f/g_{i}$ with
$i=1,2$. The minimal resolution of $fg_{i}$ is obtained after 4 point blow-
ups. Denote $E_{i}$ with $i=1,2,3,4$ the exceptional divisors, $E_{5}$ (resp.
$E_{6}$) the strict transform of $f$ (resp. of $g_{i}$). Notice that $E_{5}$
intersects $E_{4}$, and $E_{6}$ intersects $E_{1}$ if $i=1$ or $E_{2}$ if
$i=2$. Two additional point blow-ups, with exceptional divisors $E_{7}$ and
$E_{8}$ are needed to construct the resolution of $f/g_{i}$. The following
figures and tables give the corresponding dual resolution graphs and
resolution data

$E_{8}$$E_{7}$$E_{1}$$E_{3}$$E_{4}$$E_{2}$$E_{6}$$E_{5}$$E_{1}$$E_{3}$$E_{4}$$E_{2}$$E_{7}$$E_{8}$$E_{9}$$E_{5}$$E_{6}$

${\small\begin{array}[]{c|cccccccc}&E_{1}&E_{2}&E_{3}&E_{4}&E_{5}&E_{6}&E_{7}&E_{8}\\\
\hline\cr N_{f,i}&3&5&9&15&1&0&3&3\\\ N_{g_{1},i}&1&1&2&3&0&1&2&3\\\
k_{i}&1&2&4&7&0&0&2&3\\\
\end{array}\qquad\begin{array}[]{c|ccccccccc}&E_{1}&E_{2}&E_{3}&E_{4}&E_{5}&E_{6}&E_{7}&E_{8}&E_{9}\\\
\hline\cr N_{f,i}&3&5&9&15&1&0&5&5&5\\\ N_{g_{2},i}&1&2&3&5&0&1&3&4&5\\\
k_{i}&1&2&4&7&0&0&3&4&5\\\ \end{array}}$

The set of jumping numbers of $f/g_{1}$ is
$\left\\{\frac{8}{12},\frac{11}{12},1,\frac{23}{12}\right\\}\cup\mathbb{Z}_{\geq
2}$, the set of jumping numbers of $f/g_{2}$ is
$\left\\{\frac{8}{10}\right\\}\cup\mathbb{Z}_{>0}$ and the multiplier ideals
of $f/g_{1}$ and $f/g_{2}$ are

${\footnotesize{\mathcal{J}}\left(\left(\frac{f}{g_{1}}\right)^{\lambda}\right)=\begin{cases}1&0<\lambda<8/12,\\\
(x,y)&8/12\leq\lambda<11/12,\\\ (x^{2},y)&11/12\leq\lambda<1,\\\
f&1\leq\lambda<23/12,\\\ (x,y)f&23/12\leq\lambda<2\\\
f^{n}&n\leq\lambda<n+1,\\\ &\quad\hbox{ and }n\in\mathbb{Z}_{\geq
2}.\end{cases}\quad\hbox{ and
}\quad{\mathcal{J}}\left(\left(\frac{f}{g_{2}}\right)^{\lambda}\right)=\begin{cases}1&0<\lambda<8/10,\\\
(x,y)&8/10\leq\lambda<1,\\\ f^{n}&n\leq\lambda<n+1,\\\ &\quad\hbox{ and
}n\in\mathbb{Z}_{\geq 1}.\end{cases}}$

For comparison’s sake, recall that the set of jumping numbers of $f$ is
$\left\\{\frac{8}{15},\frac{11}{15},\frac{13}{15},\frac{14}{15}\right\\}+\mathbb{Z}_{\geq
0}$, and the multiplier ideals of $f$ are

${\small{\mathcal{J}}(f^{\lambda})=\begin{cases}f^{n}&n<\lambda<8/15+n,\hbox{
and }\lambda\neq 0,\\\ (x,y)f^{n}&8/15+n\leq\lambda<11/15+n,\\\
(x^{2},y)f^{n}&11/15+n\leq\lambda<13/15+n,\\\
(x,y)^{2}f^{n}&13/15+n\leq\lambda<14/15+n,\\\
(x^{3},xy,y^{2})f^{n}&14/15+n\leq\lambda<1+n.\end{cases}}$

## Acknowledgments

We thank Nero Budur for pointing out the phenomenon described in Remark 4.11.

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