††thanks: This work has been partly supported by JSPS Grant-in-Aid for
Scientific Research (C) (Nos 18K03214,18K03320).

# Methods for computing $b$-functions associated
with $\mu$-constant deformations
– Case of inner modality 2 –

Katsusuke Nabeshima  Graduate School of Technology, Industrial and Social
Sciences, Tokushima University,
2-1, Minamijosanjima, Tokushima, JAPAN<EMAIL_ADDRESS>Shinichi
Tajima  Graduate School of Science and Technology, Niigata University,
8050, Ikarashi 2-no-cho, Nishi-ku Niigata, JAPAN
<EMAIL_ADDRESS>
###### Abstract.

New methods for computing parametric local $b$-functions are introduced for
$\mu$-constant deformations of semi-weighted homogeneous singularities. The
keys of the methods are comprehensive Gröbner systems in Poincaré-Birkhoff-
Witt algebra and holonomic ${\mathcal{D}}$-modules. It is shown that the use
of semi-weighted homogeneity reduces the computational complexity of
$b$-functions associated with $\mu$-constant deformations. In the case of
inner modality 2, local $b$-functions associated with $\mu$-constant
deformations are obtained by the resulting method and given the list of
parametric local $b$-functions.

###### Key words and phrases:

$b$-function, $\mu$-constant deformations, comprehensive Gröbner system, local
cohomology, ${\mathcal{D}}$-modules

###### 1991 Mathematics Subject Classification:

Primary 13P10; Secondary 14H20

##  1\. Introduction

We introduce a new algorithm for computing local $b$-functions of semi-
weighted homogeneous polynomials, and study local $b$-functions of
$\mu$-constant deformation of inner modality 2 singularities.

A $b$-function is an analytic invariant of a hypersurface singularity. In
1997, T. Oaku gave an algorithm for computing $b$-functions [20]. After that
many researcher have improved the algorithm in the context of symbolic
computation. However, as the computational complexity of the existing
algorithms is still quite high, it is difficult to obtain $b$-functions in
realistic time in many cases. The bottleneck lies in computing non-commutative
Gröbner bases.

In a pioneering paper [22] published in 1978, T. Yano investigated
$b$-functions and already considered $b$-functions associated with a
$\mu$-constant deformations of singularities. He noticed that some roots of
$b$-functions are not stable and change these values discontinuously under
$\mu$-constant deformations. Later, in 1980’s, M. Kato and P. Cassou Nogués
explicitly computed $b$-functions associated with a $\mu$-constant deformation
for some cases in their seminal papers [3, 4, 8, 9].

In this paper, we consider $b$-functions of semi-weighted homogeneous
singularities in the context of computational algebraic analysis. Upon using
properties of semi-weighted homogeneous singularities, we construct an
effective method for computing $b$-functions. Notably, the resulting algorithm
explicitly compute parameter dependency of $b$-functions associated with
$\mu$-constant deformations. The keys of our approach are comprehensive
Gröbner systems in Poincaré-Birkhoff-Witt algebra and holonomic
${\mathcal{D}}$-modules. We show that the use of semi-weighted homogeneity
allows us to design an effective algorithm for computing $b$-functions
associated with $\mu$-constant deformations, which avoid the use of Gröbner
bases computation in an elimination step. As an application, we study inner
modality 2 singularities.

This paper is organized as follows. In section 2, we recall the notion of
semi-weighted homogeneity and give a list of semi-weighted homogeneous
polynomials whose inner modality is equal to two, the target singularities of
the present paper. In section 3, we briefly recall a comprehensive Gröbner
system, a method of computing parametric $b$-functions and review our previous
results reported in [17]. In section 4, we present an algorithm for computing
algebraic local cohomology solutions of a holonomic ${\mathcal{D}}$-module. In
section 5, we introduce a new algorithm for computing $b$-functions of semi-
weighted homogeneous polynomials. In section 6, we give a list of
$b$-functions associated with a $\mu$-constant deformation of inner modality 2
singularities.

Throughout this paper, we use the notation $x$ as the abbreviation of $n$
variables $x_{1},\ldots,x_{n}$, $\operatorname{\mathbb{Q}}$ as the field of
rational numbers and $\operatorname{\mathbb{C}}$ as the field of complex
numbers. The set of natural numbers $\operatorname{\mathbb{N}}$ includes zero.
For elements $p_{1},p_{2},\ldots,p_{r}$ in a ring $R$, let $\operatorname{{\it
Id}}(p_{1},p_{2},\ldots,p_{r})$ denote the ideal in $R$ generated by
$p_{1},p_{2},\ldots,p_{r}$.

##  2\. The list of inner modality 2 singularities

Let ${\bf w}=(w_{1},w_{2},\ldots,w_{n})\in\operatorname{\mathbb{N}}^{n}$,
$\alpha=(a_{1},a_{2},\ldots,a_{n})\in\operatorname{\mathbb{N}}^{n}$ and
$x^{\alpha}=x_{1}^{a_{1}}x_{2}^{a_{2}}\cdots
x_{n}^{a_{n}}\in\operatorname{\mathbb{C}}[x]$ . Let $|x^{\alpha}|_{{\bf w}}$
denote the weighted degree $\displaystyle\sum_{i=1}^{n}w_{i}a_{i}$ of the
monomial $x^{\alpha}$.

######  Definition 2.1.

  1. (1)

A non-zero polynomial $f\in\operatorname{\mathbb{C}}[x]$ is called weighted
homogeneous of type $(d;{\bf w})$ if all monomials of $f$ have the same
weighted degree $d$ w.r.t. ${\bf w}$ where $d\in\operatorname{\mathbb{N}}$.

  2. (2)

The polynomial $f$ is called semi-weighted homogeneous of type $(d;{\bf w})$
if $f$ is of the form $f=f_{0}+g$ where $f_{0}$ is a weighted homogeneous
polynomial of type $(d;{\bf w})$ with an isolated singularity at the origin
$O$ in $\operatorname{\mathbb{C}}^{n}$, and $f=f_{0}$ or $\text{ord}_{{\bf
w}}(f-f_{0})>d$ where $\text{ord}_{{\bf w}}(f)=\text{min}\\{|x^{\alpha}|_{{\bf
w}}:x^{\alpha}\text{ is a monomial of }f\\}$. A monomial of $g$ is called an
upper of $f_{0}$.

Table 1. List of inner modality 2 singularities type | weighted homo. $f_{0}$ | upper monomials | note
---|---|---|---
$E_{18}$ | $x^{3}+y^{10}$ | $xy^{7},xy^{8}$ |
$E_{19}$ | $x^{3}+xy^{7}$ | $y^{11},y^{12}$ |
$E_{20}$ | $x^{3}+y^{11}$ | $xy^{8},xy^{9}$ |
$W_{17}$ | $x^{4}+xy^{5}$ | $y^{7},y^{8}$ |
$W_{18}$ | $x^{4}+y^{7}$ | $x^{2}y^{4},x^{2}y^{5}$ |
$Z_{17}$ | $x^{3}y+y^{8}$ | $xy^{6},xy^{7}$ |
$Z_{18}$ | $x^{3}y+xy^{6}$ | $y^{9},y^{10}$ |
$Z_{19}$ | $x^{3}y+y^{9}$ | $xy^{7},xy^{8}$ |
$Q_{16}$ | $x^{3}+yz^{2}+y^{7}$ | $xy^{5},xz^{2}$ |
$Q_{17}$ | $x^{3}+yz^{2}+xy^{5}$ | $y^{8},y^{9}$ |
$Q_{18}$ | $x^{3}+yz^{2}+y^{8}$ | $xy^{6},xz^{2}$ |
$S_{16}$ | $x^{2}z+yz^{2}+xy^{4}$ | $y^{6},z^{3}$ |
$S_{17}$ | $x^{2}z+yz^{2}+y^{6}$ | $y^{4}z,z^{3}$ |
$U_{16}$ | $x^{3}+xz^{2}+y^{5}$ | $y^{2}z^{2},y^{3}z^{2}$ |
$J_{16}$ | $x^{3}+y^{9}+u_{1}x^{2}y^{3}$ | $y^{10}$ | $4u_{1}^{3}+27\neq 0$
$W_{15}$ | $x^{4}+y^{6}+u_{1}x^{2}y^{3}$ | $y^{7}$ | $u_{1}^{2}-4\neq 0$
$Z_{15}$ | $x^{3}y+y^{7}+u_{1}x^{2}y^{3}$ | $y^{8}$ | $4u_{1}^{3}+27\neq 0$
$Q_{14}$ | $x^{3}+yz^{2}+u_{1}x^{2}y^{2}+xy^{4}$ | $y^{7}$ | $u_{1}^{2}-4\neq 0$
$S_{14}$ | $x^{2}z+yz^{2}+y^{5}+u_{1}y^{3}z$ | $z^{3}$ | $u_{1}^{2}-4\neq 0$
$U_{14}$ | $x^{3}+xz^{2}+u_{1}xy^{3}+y^{3}z$ | $yz^{3}$ | $u_{1}^{2}+1\neq 0$

Note that the Milnor number at the origin of $f=f_{0}+g$ and that of $f_{0}$
are same, and the embedded topological type of $f=f_{0}+g$ singularity and
that of $f_{0}$ are same [10, 24]. Accordingly, $f=f_{0}+g$ is called a
$\mu$-constant deformation of $f_{0}.$

In [23], Yoshinaga and Suzuki gave lists of normal forms of quasihomogeneous
(weighted homogeneous) functions with inner modality $\leq 4$. Table 1 quoted
from [23] is the list of inner modality 2 singularities.

For example, $E_{18}$ in the table means the following.

  1. (i)

$f_{0}(x,y)=x^{3}+y^{10}$ is a weighted homogeneous polynomial (of type
$(30,(10,3))$.

  2. (ii)

The Milnor number of $f_{0}(x,y)$ at the origin is equal to $18.$

  3. (iii)

$f(x,y)=f_{0}(x,y)+c_{(1,7)}xy^{7}+c_{(1,8)}xy^{8}$ is a semi-weighted
homogeneous polynomial (of type (30, (10,3)) where
$c_{(1,7)},c_{(1,8)}\in\operatorname{\mathbb{C}}$.

##  3\. Comprehensive Gröbner system approach

Here we briefly recall a comprehensive Gröbner system approach to compute
parametric $b$-functions and review our previous results reported in [17] on
the computation of $b$-functions associated with $\mu$-constant deformations.

Let $D=\operatorname{\mathbb{C}}[x,\partial]$ denote the Weyl algebra, the
ring of linear partial differential operators with coefficients in
$\mathbb{C}[x]$, where $\partial=\\{\partial_{1},\ldots,\partial_{n}\\}$,
$\partial_{i}=\frac{\partial}{\partial x_{i}}$, i.e.,

$\displaystyle
D=\left\\{{\sum_{\beta\in\operatorname{\mathbb{N}}^{n}}h_{\beta}(x)\partial^{\beta}}\left|\,\vphantom{\sum_{\beta\in\operatorname{\mathbb{N}}^{n}}h_{\beta}(x)\partial^{\beta}}{h_{\beta}(x)\in\operatorname{\mathbb{C}}[x]}\right.\right\\}.$

Throughout the paper we assume that a linear partial differential operator is
always represented in the canonical form : each power product of a partial
differential operator is written as $x^{\alpha}{\partial}^{\beta}$ where
$\alpha,\beta\in\operatorname{\mathbb{N}}^{n}$.

Let $u=\\{u_{1},\ldots,u_{m}\\}$ be variables such that $u\cap x=\emptyset$,
$D[u]$ a ring of partial differential operators with coefficients in a
polynomial ring $(\operatorname{\mathbb{C}}[u])[x]$, i.e.,

$D[u]=\left\\{{\sum_{\beta\in\operatorname{\mathbb{N}}^{n}}h_{\beta}(u,x)\partial^{\beta}}\left|\,\vphantom{\sum_{\beta\in\operatorname{\mathbb{N}}^{n}}h_{\beta}(u,x)\partial^{\beta}}{h_{\beta}(u,x)\in(\operatorname{\mathbb{C}}[u])[x]}\right.\right\\}.$

Let $D\langle s,\partial_{t}\rangle$ (or $D[u]\langle s,\partial_{t}\rangle$)
denote the Poincaré-Birkhoff-Witt (PBW) algebra
$D\otimes_{\operatorname{\mathbb{C}}}\operatorname{\mathbb{C}}[s,\partial_{t}]$
(or
$D[u]\otimes_{\operatorname{\mathbb{C}}}\operatorname{\mathbb{C}}[s,\partial_{t}]$)
with a non-commutative relation $\partial_{t}s=s\partial_{t}-\partial_{t}$ and
commutative relations
$\partial_{t}x_{i}=x_{i}\partial_{t},\partial_{i}s=s\partial_{i},sx_{i}=x_{i}s,\partial_{t}\partial_{i}=\partial_{i}\partial_{t}$
($1\leq i\leq n$). There exist algorithms and implementations to compute
Gröbner bases of given ideals in the non-commutative rings $D$ and $D\langle
s,\partial_{t}\rangle$ [5, 11].

For $g_{1},\ldots,g_{r}\in\operatorname{\mathbb{C}}[u]$,
$\operatorname{\mathbb{V}}(g_{1},\ldots,g_{r})\subseteq\operatorname{\mathbb{C}}^{m}$
denotes the affine variety of $g_{1},\ldots,g_{r}$, i.e.,
$\operatorname{\mathbb{V}}(g_{1},\ldots,g_{r})=\\{\bar{u}\in\operatorname{\mathbb{C}}^{m}|g_{1}(\bar{u})=\cdots=g_{r}(\bar{u})=0\\}$.
For
$g_{1},\ldots,g_{r},g^{\prime}_{1},\ldots,g^{\prime}_{r^{\prime}}\in\operatorname{\mathbb{C}}[t]$,
we call an algebraic constructible set
$\operatorname{\mathbb{V}}(g_{1},\ldots,g_{r})\backslash\operatorname{\mathbb{V}}(g^{\prime}_{1},\ldots,g^{\prime}_{r^{\prime}})\subseteq\operatorname{\mathbb{C}}^{m}$
a stratum. Notations
$\operatorname{\mathbb{U}}_{1},\operatorname{\mathbb{U}}_{2},$
$\ldots,\operatorname{\mathbb{U}}_{\ell}$ are frequently used to represent
strata.

For every $\bar{u}\in\operatorname{\mathbb{C}}^{m}$, the canonical
specialization homomorphism $\sigma_{\bar{u}}:D[u]\langle
s,\partial_{t}\rangle\rightarrow D\langle s,\partial_{t}\rangle$ (or
$D[u]\rightarrow D$) is defined as the map that substitutes $u$ by $\bar{u}$
in $p(u,x,\partial,s,$ $\partial_{t})\in D[u]\langle s,\partial_{t}\rangle$.
The image $\sigma_{\bar{u}}$ of a set $F$ is denoted by
$\sigma_{\bar{u}}(F)=\\{\sigma_{\bar{u}}(p)|p\in F\\}\subset D\langle
s,\partial_{t}\rangle$. A symbol
$\operatorname{Mono}(x\cup\partial\cup\\{s,\partial_{t}\\})$ is the set of
monomials of $x\cup\partial\cup\\{s,\partial_{s}\\}$.

The main tool to compute $b$-functions associated with $\mu$-constant
deformations, is a comprehensive Gröbner system in the PBW algebra. We adopt
the following as a definition of comprehensive Gröbner systems.

######  Definition 3.1 (CGS).

Let $\succ$ be a monomial order on
$\operatorname{Mono}(x\cup\partial\cup\\{s,\partial_{t}\\})$. Let $F$ be a
subset of $D[u]\langle s,\partial_{t}\rangle$,
$\operatorname{\mathbb{U}}_{1},\operatorname{\mathbb{U}}_{2},\ldots,\operatorname{\mathbb{U}}_{\ell}$
strata in $\operatorname{\mathbb{C}}^{m}$ and $G_{1},\ldots,G_{\ell}$ subsets
in $D[u]\langle s,\partial_{t}\rangle$. If a finite set
${\mathcal{G}}=\\{(\operatorname{\mathbb{U}}_{1},G_{1}),\ldots,(\operatorname{\mathbb{U}}_{\ell},G_{\ell})\\}$
of pairs satisfies properties such that

  1. (1)

$\operatorname{\mathbb{U}}_{i}\neq\emptyset$ and
$\operatorname{\mathbb{U}}_{i}\cap\operatorname{\mathbb{U}}_{j}=\emptyset$ for
$1\leq i\neq j\leq\ell$,

  2. (2)

for all $\bar{u}\in\operatorname{\mathbb{U}}_{i}$, $\sigma_{\bar{u}}(G_{i})$
is a minimal Gröbner basis of $\operatorname{{\it Id}}(\sigma_{\bar{u}}(F))$
w.r.t. $\succ$ in $D\langle s,\partial_{t}\rangle$, and

  3. (3)

for all $\bar{u}\in\operatorname{\mathbb{U}}_{i}$ and $p\in G_{i}$,
$\sigma_{\bar{u}}(\operatorname{hc}(p))\neq 0$ where $\operatorname{hc}(p)$ is
the head coefficient of $p$ in $\operatorname{\mathbb{C}}[u]$,

${\mathcal{G}}$ is called a comprehensive Gröbner system (CGS) on
$\operatorname{\mathbb{U}}_{1}\cup\cdots\cup\operatorname{\mathbb{U}}_{\ell}$
for $\operatorname{{\it Id}}(F)$ w.r.t. $\succ$. We simply say that
${\mathcal{G}}$ is a comprehensive Gröbner system for $\operatorname{{\it
Id}}(F)$ if
$\operatorname{\mathbb{U}}_{1}\cup\cdots\cup\operatorname{\mathbb{U}}_{\ell}=\operatorname{\mathbb{C}}^{m}$.

In our previous papers [14, 15], algorithms and implementations for computing
comprehensive Gröbner systems in PBW algebras are introduced.

###  3.1. Global $b$-functions

Let $f$ be a non-constant polynomial in $\mathbb{C}[x]$. Then, the
annihilating ideal of $f^{s}$ is

$\text{Ann}(f^{s})=\\{p\in D[s]|pf^{s}=0\\}$

where $s$ is an indeterminate, and

$\displaystyle
D[s]=\left\\{{\sum_{k\in\operatorname{\mathbb{N}},\beta\in\operatorname{\mathbb{N}}^{n}}h_{k,\beta}(x)s^{k}\partial^{\beta}}\left|\,\vphantom{\sum_{k\in\operatorname{\mathbb{N}},\beta\in\operatorname{\mathbb{N}}^{n}}h_{k,\beta}(x)s^{k}\partial^{\beta}}{h_{k,\beta}(x)\in\operatorname{\mathbb{C}}[x]}\right.\right\\}.$

Consider the following left ideal $I$ in the PBW algebra $D\langle
s,\partial_{t}\rangle$.

$I=\operatorname{{\it
Id}}\left(f\cdot\partial_{t}+s,\partial_{1}+\partial_{t}\cdot\frac{\partial
f}{\partial x_{1}},\partial_{2}+\partial_{t}\cdot\frac{\partial f}{\partial
x_{2}},\ldots,\partial_{n}+\partial_{t}\cdot\frac{\partial f}{\partial
x_{n}}\right).$

Briançon and Maisonobe show in [2] that $\text{Ann}(f^{s})=I\cap D[s].$ Thus,
a basis of the ideal $\text{Ann}(f^{s})$ can be obtained by the Gröbner basis
computation of $I$ w.r.t. an elimination order for $\partial_{t}$.

The global $b$-function or the global Bernstein-Sato polynomial of $f$ is
defined as the monic generator $b_{f}(s)$ of
$(\text{Ann}(f^{s})+\operatorname{{\it Id}}(f))\cap\mathbb{C}[s]$ where
$\operatorname{{\it Id}}(f)$ is the ideal generated by $f$. It is known that
the $b$-function of $f$ always has $s+1$ as a factor and has a form
$(s+1)\tilde{b}_{f}(s)$, where $\tilde{b}_{f}(s)\in\mathbb{C}[s]$. The
polynomial $\tilde{b}_{f}(s)$ is called the (global) reduced $b$-function of
$f$. The reduced $b$-function $\tilde{b}_{f}(s)$ can be obtained by computing
a Gröbner basis of $\text{Ann}(f^{s})+\operatorname{{\it Id}}(f,\frac{\partial
f}{\partial x_{1}},\ldots,\frac{\partial f}{\partial x_{n}})$ w.r.t. a block
order $\\{x,\partial\\}\gg s$.

Let $f$ be a parametric polynomial in $(\mathbb{C}[u])[x]$ where $u$ are
regarded as parameters. As mentioned previously, a CGS of the
$\text{Ann}(f^{s})$ is computable by the algorithm for computing CGS’s in PBW
algebras. Accordingly, global $b$-functions with parameters are computable by
using CGS’s of the ideals $\text{Ann}(f^{s})+\text{Id}(f)$ and
$\text{Ann}(f^{s})+\operatorname{{\it Id}}(f,\frac{\partial f}{\partial
x_{1}},\ldots,\frac{\partial f}{\partial x_{n}})$. We refer the reader to [14]
and [15] for details.

Algorithm 1. (Global $b$-functions)

Input: $\operatorname{\mathbb{U}}\subseteq\operatorname{\mathbb{C}}^{m}$,
$f\in({\operatorname{\mathbb{C}}}[u])[x]$: $f$ is a non-constant polynomial
with parameter $u$,
Output:
${\mathcal{G}}=\\{(\operatorname{\mathbb{U}}_{1},\tilde{b}_{1}(s)),\ldots,(\operatorname{\mathbb{U}}_{\ell},\tilde{b}_{\ell}(s))\\}$:
for each $i\in\\{1,\ldots,\ell\\}$ and
$\forall\bar{u}\in\operatorname{\mathbb{U}}_{i}$, $\tilde{b}_{i}(s)$ is the
reduced $b$-function of $\sigma_{\bar{u}}(f)$ and
$\operatorname{\mathbb{U}}=\bigcup_{i=1}^{\ell}\operatorname{\mathbb{U}}_{i}$.
BEGIN
${\mathcal{G}}\leftarrow\emptyset$; $J\leftarrow\\{\frac{\partial f}{\partial
x_{1}},\ldots,\frac{\partial f}{\partial x_{n}}\\}$;
$I\leftarrow\operatorname{{\it
Id}}(f\cdot\partial_{t}+s,\partial_{1}+\partial_{t}\cdot\frac{\partial
f}{\partial x_{1}},\partial_{2}+\partial_{t}\cdot\frac{\partial f}{\partial
x_{2}},\ldots,\partial_{n}+\partial_{t}\cdot\frac{\partial f}{\partial
x_{n}})$;
${\mathcal{P}}\leftarrow$ Compute a CGS of $I$ w.r.t. an elimination order for
$\partial_{t}$ on $\operatorname{\mathbb{U}}$;
while ${\mathcal{P}}\neq\emptyset$ do
Select $(\operatorname{\mathbb{U}}^{\prime},P)$ from ${\mathcal{P}}$;
${\mathcal{P}}\leftarrow{\mathcal{P}}\backslash\\{(\operatorname{\mathbb{U}}^{\prime},P)\\}$;
${\mathcal{B}}\leftarrow$ Compute a CGS of $\operatorname{{\it
Id}}(J\cup(P\cap D[s]))$ on $\operatorname{\mathbb{U}}^{\prime}$ w.r.t. a
block order $\\{x,\partial\\}\gg s$; $(\ast 1)$
while ${\mathcal{B}}\neq\emptyset$ do
Select $(\operatorname{\mathbb{U}}^{\prime\prime},B)$ from ${\mathcal{B}}$;
${\mathcal{B}}\leftarrow{\mathcal{B}}\backslash\\{(\operatorname{\mathbb{U}}^{\prime\prime},B)\\}$;
${\mathcal{G}}\leftarrow{\mathcal{G}}\cup\\{(\operatorname{\mathbb{U}}^{\prime\prime},B\cap\operatorname{\mathbb{C}}[s])\\}$;
end-while
end-while
return ${\mathcal{G}}$;
END

Note that the rationality of the roots of $b$-functions has been shown by [7].
Hence, the $b$-functions can be factorized into linear factors over
$\operatorname{\mathbb{Q}}$.

######  Example 1.

Let us consider $U_{16}$ singularity
$f=x^{3}+xz^{2}+y^{5}+u_{1}y^{2}z^{2}+u_{2}y^{3}z^{2}$ where $u_{1},u_{2}$ are
parameters. As $f$ is a semi-weighted homogeneous, the Milnor number of the
singularity at the origin is 16. Set
$b_{\text{st}}(s)=(s+\frac{13}{15})(s+\frac{16}{15})(s+\frac{18}{15})(s+\frac{19}{15})(s+\frac{21}{15})(s+\frac{22}{15})(s+\frac{23}{15})(s+\frac{24}{15})(s+\frac{26}{15})(s+\frac{27}{15}).$
Then the reduced $b$-functions of $f$ is the following.

  1. $\bullet$

If $(u_{1},u_{2})$ belongs to
$\operatorname{\mathbb{C}}^{2}\backslash\operatorname{\mathbb{V}}(u_{1}(27u_{1}^{4}+256u_{2}))$,
then $\tilde{b}_{f}(s)=b_{\text{st}}(s)(s+\frac{14}{15})(s+\frac{17}{15})$.

  2. $\bullet$

If $(u_{1},u_{2})$ belongs to
$\operatorname{\mathbb{V}}(27u_{1}^{4}+256u_{2})\backslash\operatorname{\mathbb{V}}(u_{1},u_{2})$,
then
$\tilde{b}_{f}(s)=b_{\text{st}}(s)(s+\frac{3}{2})(s+\frac{14}{15})(s+\frac{17}{15})$.

  3. $\bullet$

If $(u_{1},u_{2})$ belongs to
$\operatorname{\mathbb{V}}(u_{1})\backslash\operatorname{\mathbb{V}}(u_{1},u_{2})$,
then $\tilde{b}_{f}(s)=b_{\text{st}}(s)(s+\frac{17}{15})(s+\frac{29}{15})$.

  4. $\bullet$

If $(u_{1},u_{2})$ belongs to $\operatorname{\mathbb{V}}(u_{1},u_{2})$, then
$\tilde{b}_{f}(s)=b_{\text{st}}(s)(s+\frac{29}{15})(s+\frac{32}{15})$.

Note that if $(u_{1},u_{2})$ belongs to
$\operatorname{\mathbb{V}}(27u_{1}^{4}+256u_{2})\backslash\operatorname{\mathbb{V}}(u_{1},u_{2})$,
then $f$ has three isolated singularities $(0,0,0)$ and
$\displaystyle\left(\frac{3u_{1}^{3}}{64u_{2}^{2}},\frac{u_{1}}{4u_{2}},\pm\sqrt{\frac{-16}{27u_{1}^{2}u_{2}^{2}}}\right)$
in $\operatorname{\mathbb{C}}^{3}$. In other cases, $f$ has one isolated
singularity $(0,0,0)$.

###  3.2. Local $b$-functions

Let

$\displaystyle{\mathcal{D}}[s]=\left\\{{\sum_{k\in\operatorname{\mathbb{N}},\beta\in\operatorname{\mathbb{N}}^{n}}h_{k,\beta}(x)s^{k}\partial^{\beta}}\left|\,\vphantom{\sum_{k\in\operatorname{\mathbb{N}},\beta\in\operatorname{\mathbb{N}}^{n}}h_{k,\beta}(x)s^{k}\partial^{\beta}}{h_{k,\beta}(x)\in\operatorname{\mathbb{C}}[x]_{q}}\right.\right\\}$

where $\operatorname{\mathbb{C}}[x]_{q}=\\{g_{1}(x)/g_{2}(x)|\
g_{1}(x),g_{2}(x)\in\operatorname{\mathbb{C}}[x],g_{2}(q)\neq 0\\}$ the
localization of $\operatorname{\mathbb{C}}[x]$ at
$q\in\operatorname{\mathbb{C}}^{n}$. The local $b$-function of a non-constant
polynomial $f\in\operatorname{\mathbb{C}}[x]$ at $q$ is defined as the monic
polynomial $b_{f,q}(s)$ of the minimal degree for $p\in{\mathcal{D}}[s]$ and
$b_{f,q}(s)\in\operatorname{\mathbb{C}}[s]$ satisfying $p\cdot
f^{s+1}=a(x)b_{f,q}(s)\cdot f^{s}$ where
$a(x)\in\operatorname{\mathbb{C}}[x]_{q}$. The reduced $b$-function of $f$ at
$q$, written as $\tilde{b}_{f,q}(s)$, is $b_{f,q}(s)/(s+1)$.

In [13], Mebkhout and Narváez-Macarro show the fact

$\tilde{b}_{f}(s)=\text{LCM}(\tilde{b}_{f,q}(s)|q\in\text{Sing}(f))$

where $\text{Sing}(f)$ is the singular locus of
$\operatorname{\mathbb{V}}(f)$, i.e.,
$\text{Sing}(f)=\operatorname{\mathbb{V}}(f,\frac{\partial f}{\partial
x_{1}},\ldots,\frac{\partial f}{\partial x_{n}})$.

We borrow from [22] the following theorem.

######  Theorem 3.2.

Let $f\in\operatorname{\mathbb{C}}[x]$, $\gamma\in\operatorname{\mathbb{Q}}$
and set

$M_{(\gamma,f)}=D[s]/(\text{Ann}(f^{s})+\operatorname{{\it
Id}}(f,\frac{\partial f}{\partial x_{1}},\ldots,\frac{\partial f}{\partial
x_{n}})+\operatorname{{\it Id}}(s-\gamma)).$

Then, if $\tilde{b}_{f}(\gamma)\neq 0$, then $M_{(\gamma,f)}=\\{0\\}$, and if
$\tilde{b}_{f}(\gamma)=0$, then $M_{\gamma}$ is a holonomic $D$-module and
$\text{supp}(M_{(\gamma,f)})\subseteq\text{Sing}(f)$ where
$\text{supp}(M_{(\gamma,f)})$ is the support of $M_{(\gamma,f)}$.

######  Lemma 3.3.

Using the same notation as in Theorem 3.2, let $\gamma$ be a root of
$\tilde{b}_{f,q}(s)=0$. Then, $q\in\text{supp}(M_{(\gamma,f)})$.

Algorithms and implementations for computing $M_{(\gamma,f)}$ and
$\text{supp}(M_{(\gamma,f)})$, have been already introduced in [11] and in
[14, 15, 20].

After here, we consider local $b$-functions of semi-weighted homogeneous
polynomials at $O\in\operatorname{\mathbb{C}}^{n}$. For a semi-weighted
homogeneous polynomial, the following property is known.

######  Theorem 3.4 (C.2.1.6. [1]).

For a semi-weighted homogeneous polynomial $f\in\operatorname{\mathbb{C}}[x]$,
the local $b$-function $\tilde{b}_{f,0}(s)$ is square-free, i.e.,
$\tilde{b}_{f,0}(s)=0$ has no multiple roots.

Note that, there is a possibility that the global $b$-function of a semi-
weighted homogeneous polynomial has multiple roots, whereas as Theorem 3.4
says, the local $b$-function of a semi-weighted homogeneous polynomial has no
multiple roots.

We turn to parametric cases. Let $f$ be a semi-weighted homogeneous polynomial
in $(\operatorname{\mathbb{C}}[u])[x]$. Then, by utilizing Lemma 3.3 and
Theorem 3.4, we are able to construct an algorithm for computing local
$b$-functions of $f$ at $O\in\operatorname{\mathbb{C}}^{n}$.

Algorithm 2. (Local $b$-functions at $O$)

Input: $\operatorname{\mathbb{U}}\subseteq\operatorname{\mathbb{C}}^{m}$,
$f=f_{0}+g\in(\operatorname{\mathbb{C}}[u])[x]$:
$\forall\bar{u}\in\operatorname{\mathbb{U}}$, $\sigma_{\bar{u}}(f_{0})$ is a
weighted homogeneous polynomial with an isolated singularity at $O$ and
$\sigma_{\bar{u}}(f)$ is a semi-weighted homogeneous polynomial,
Output:
${\mathcal{G}}_{0}=\\{(\operatorname{\mathbb{U}}_{1},\tilde{b}_{1}(s)),\ldots,(\operatorname{\mathbb{U}}_{\ell},\tilde{b}_{\ell}(s))\\}$:
for each $i\in\\{1,\ldots,\ell\\}$ and
$\forall\bar{u}\in\operatorname{\mathbb{U}}_{i}$, $\tilde{b}_{i}(s)$ is the
local reduced $b$-function of $\sigma_{\bar{u}}(f)$ at $O$ (i.e.,
$\tilde{b}_{i}(s)=\tilde{b}_{\sigma_{\bar{u}}(f),0}(s)$) and
$\operatorname{\mathbb{U}}=\bigcup_{i=1}^{\ell}\operatorname{\mathbb{U}}_{i}$.
BEGIN
${\mathcal{G}}_{0}\leftarrow\emptyset$; ${\mathcal{G}}\leftarrow$ Execute
Algorithm 1 on $\operatorname{\mathbb{U}}$;
while ${\mathcal{G}}\neq\emptyset$ do
Select $(\operatorname{\mathbb{U}}^{\prime},b(s))$ from ${\mathcal{G}}$;
${\mathcal{G}}\leftarrow{\mathcal{G}}\backslash\\{(\operatorname{\mathbb{U}}^{\prime},b(s))\\}$;
$E\leftarrow$ Compute all roots of $b(s)=0$;
$\tilde{b}\leftarrow 1;$
while $E\neq\emptyset$ do
Select $\gamma$ from $E$; $E\leftarrow E\backslash\\{\gamma\\}$;
if $O\in\text{supp}(M_{(\gamma,f)})$ then
$\tilde{b}\leftarrow\tilde{b}\cdot(s-\gamma)$;
end-if
end-while
${\mathcal{G}}_{0}\leftarrow{\mathcal{G}}_{0}\cup\\{(\operatorname{\mathbb{U}}^{\prime},\tilde{b})\\}$;
end-while
return ${\mathcal{G}}_{0}$;
END

######  Example 2.

Let us consider Example 1, again. The Milnor number $\mu$ of the singularity
$x^{3}+xz^{2}+y^{5}=0$, at the origin $O$, is 16, and the $\mu$-constant
deformation is given by $f=x^{3}+xz^{2}+y^{5}+u_{1}y^{2}z^{2}+u_{2}y^{3}z^{2}$
where $u_{1},u_{2}$ are parameters.

According to Algorithm 2, we compute the local $b$-function
$\tilde{b}_{f,0}(s)$. We consider the case where
$(u_{1},u_{2})\in\operatorname{\mathbb{V}}(27u_{1}^{4}+256u_{2})\backslash\operatorname{\mathbb{V}}(u_{1},u_{2})$
and examine the holonomic $D$-module $M_{(-\frac{3}{2},f)}$ associated with
the factor $s+\frac{3}{2}$ of $\tilde{b}_{f}(s)$. Direct computation shows

$\text{supp}(M_{(-\frac{3}{2},f)})=\operatorname{\mathbb{V}}(64u_{2}^{2}x-3u_{1}^{3},4u_{2}y-u_{1},16u_{2}^{3}z^{2}-u_{1}^{2}).$

Since $O\notin\text{supp}(M_{(-\frac{3}{2},f)})$, $s+\frac{3}{2}$ is not a
factor of $\tilde{b}_{f,0}(s)$. The supports associated with other roots,
contain the origin $O$. Therefore, we obtain Table 2 as $\tilde{b}_{f,0}(s)$
where $b_{\text{st}}(s)$ is from Example 1.

Table 2. List of local $b$-functions stratum | (Local) $\tilde{b}_{f,0}(s)$
---|---
$\operatorname{\mathbb{C}}^{2}\backslash\operatorname{\mathbb{V}}(u_{1})$ | $b_{\text{st}}(s)(s+\frac{14}{15})(s+\frac{17}{15})$
$\operatorname{\mathbb{V}}(u_{1})\backslash\operatorname{\mathbb{V}}(u_{2})$ | $b_{\text{st}}(s)(s+\frac{17}{15})(s+\frac{29}{15})$
$\operatorname{\mathbb{V}}(u_{1},u_{2})$ | $b_{\text{st}}(s)(s+\frac{29}{15})(s+\frac{32}{15})$

As the example above shows that Algorithm 2 gives a computation method of
$b$-functions associated with $\mu$-constant deformations. We implemented
Algorithm 2 in the computer algebra system Risa/Asir [19], and we tried to
compute all problems of Table 1 for three months by using 3 computers: PC1
[OS: Linux, CPU: Xeon E3-1225, 3.2 GHz, Memory: 126 GB], PC2 [OS: Linux, CPU:
Xeon E3-1230, 3.3 GHz, Memory: 504 GB] and PC3 [OS: Windows 10, CPU: Core
i7-5930k, 3.5 GHz, Memory: 64 GB].

All bases of the annihilating ideals of $f^{s}$ in Table 1 were successfully
obtained. It turned out by this computer experiment that the cost of
computation of the part $(\ast 1)$ of Algorithm 1 (i.e., computing a CGS of
$\text{Ann}(f^{s})+\operatorname{{\it Id}}(f,\frac{\partial f}{\partial
x_{1}},\ldots,\frac{\partial f}{\partial x_{n}})$) w.r.t. an elimination
order) is quite high. Among 20 cases, our implementation could not return 12
$b$-functions within three months and output only 8 $b$-functions :
$b$-functions of $E_{18}$, $E_{20}$, $W_{18}$, $Z_{17}$, $Q_{16}$, $Q_{17}$,
$S_{17}$, $U_{16}$. We see that the direct use of Algorithm 2 is not adequate
for computing $b$-functions associated with $\mu$-constant deformations. In
order to overcome difficulties, we improve the method presented in this
section by specializing Algorithm 2 to handle semi-weighted homogeneous cases.

##  4\. Computing local cohomology solutions to a holonomic $D$-module

Here we introduce an algorithm for computing local cohomology solutions of the
holonomic $D$-module $M_{(\gamma,f)}$. The algorithm will be utilized as a key
tool in the new computation method of $b$-functions.

All local cohomology classes, in this paper, are algebraic local cohomology
classes that belong to the set defined by

$H_{[O]}^{n}(\operatorname{\mathbb{C}}[x])=\lim_{k\rightarrow\infty}\text{Ext}^{n}_{\operatorname{\mathbb{C}}[x]}(\operatorname{\mathbb{C}}[x]/\langle
x_{1},x_{2},\ldots,x_{n}\rangle^{k},\operatorname{\mathbb{C}}[x])$

where $\langle x_{1},x_{2},\ldots,x_{n}\rangle$ is the maximal ideal generated
by $x_{1},\ldots,x_{n}$. We adopt notations used in [16] to represent
algebraic local cohomology classes, namely, we represent an algebraic local
cohomology class $\sum c_{\lambda}\left[\begin{array}[]{c}1\\\
x^{\lambda+1}\end{array}\right]$ as a polynomial $\sum
c_{\lambda}\xi^{\lambda}$ where $\xi$ is the abbreviation of $n$ variables
$\xi_{1},\ldots,\xi_{n}$, $c_{\lambda}\in\operatorname{\mathbb{C}}$ and
$\lambda=(\lambda_{1},\ldots,\lambda_{n})\in\operatorname{\mathbb{N}}^{n}$.
The multiplication is defined as

$x^{\alpha}\ast\xi^{\lambda}=\left\\{\begin{array}[]{ll}\xi^{\lambda-\alpha},\
\ \ \ \ \ \ \lambda_{i}\geq\alpha_{i},i=1,\ldots,n,\\\ \\\ 0,\ \ \ \ \ \ \ \ \
\ \ \ \ \ \ \ \text{otherwise,}\end{array}\right.$

where
$\alpha=(\alpha_{1},\alpha_{2},\ldots,\alpha_{n})\in\operatorname{\mathbb{N}}^{n}$
and
$\lambda-\alpha=(\lambda_{1}-\alpha_{1},\ldots,\lambda_{n}-\alpha_{n})\in\operatorname{\mathbb{N}}^{n}$.
The partial derivative by $\frac{\partial}{\partial x_{i}}$ is defined as

$\frac{\partial}{\partial
x_{i}}\ast(\xi_{1}^{\lambda_{1}}\xi_{2}^{\lambda_{2}}\cdots\xi_{i}^{\lambda_{i}}\cdots\xi_{n}^{\lambda_{n}})=-(\lambda_{i}+1)\xi_{1}^{\lambda_{1}}\xi_{2}^{\lambda_{2}}\cdots\xi_{i}^{\lambda_{i}+1}\cdots\xi_{n}^{\lambda_{n}}.$

Let fix a monomial order $\succ$. For a given algebraic local cohomology class
of the form

$\psi=c_{\lambda}\xi^{\lambda}+\sum_{\xi^{\lambda}\succ\xi^{\lambda^{\prime}}}c_{\lambda^{\prime}}\xi^{\lambda^{\prime}},\
\ c_{\lambda}\neq 0$

we call $\xi^{\lambda}$ the head monomial, $c_{\lambda}$ the head coefficient
and $\xi^{\lambda^{\prime}}$ the lower monomials. We write the head monomial
as $\operatorname{hm}(\psi)$.

Let $f$ be a holomorphic function defined on an open neighborhood $X$ of the
origin $O$ of the $n$-dimensional complex space
$\operatorname{\mathbb{C}}^{n}$, with an isolated singularity at the origin.
Let $\gamma$ be a root of the local reduced $b$-function $\tilde{b}_{f,0}(s)$
at $O$. Let $G^{\prime}$ be a minimal Gröbner basis of
$\text{Ann}(f^{s})+\operatorname{{\it Id}}(f,\frac{\partial f}{\partial
x_{1}},\ldots,$ $\frac{\partial f}{\partial x_{n}})+\operatorname{{\it
Id}}(s-\gamma)$ w.r.t. a monomial order satisfying $\\{x,\partial\\}\gg s$ in
$D[s]$. Set $G_{(\gamma,f)}=G^{\prime}\backslash\\{s-\gamma\\}$, then
$M_{(\gamma,f)}=D/\operatorname{{\it Id}}(G_{(\gamma,f)})$. We define a set
$H_{M_{(\gamma,f)}}$ to be the set of algebraic local cohomology classes in
$H^{n}_{[O]}(\operatorname{\mathbb{C}}[x])$ that are annihilated by
$G_{(\gamma,f)}$:

$H_{M_{(\gamma,f)}}=\left\\{{\psi\in
H^{n}_{[O]}(\operatorname{\mathbb{C}}[x])}\left|\,\vphantom{\psi\in
H^{n}_{[O]}(\operatorname{\mathbb{C}}[x])}{h\ast\psi=0,\forall h\in
G_{(\gamma,f)}}\right.\right\\}.$

Since $H_{M_{(\gamma,f)}}$ is the algebraic local cohomology solution space of
the holonomic $D$-module $M_{(\gamma,f)},$ we have the following.

######  Theorem 4.1.

The set $H_{M_{(\gamma,f)}}$ is a finite dimensional vector space.

Here we introduce an algorithm for computing a basis of the vector space
$H_{M_{(\gamma,f)}}$.

######  Lemma 4.2.

Using the same notation as in above, let
$P_{0}=G_{(\gamma,f)}\cap\operatorname{\mathbb{C}}[x]$,
$F_{0}=P_{0}\cup\\{f,\frac{\partial f}{\partial x_{1}},\ldots,$
$\frac{\partial f}{\partial x_{n}}\\}\subset\operatorname{\mathbb{C}}[x]$. Set

$H_{F_{0}}=\\{\psi\in
H^{n}_{[O]}(\operatorname{\mathbb{C}}[x])|h\ast\psi=0,\forall h\in F_{0}\\}.$

Then, $H_{M_{(\gamma,f)}}\subseteq H_{F_{0}}$.

Since $P_{0}\subset G_{(\gamma,f)}$, Lemma 4.2 holds. Note that
$F_{0}\subset\operatorname{\mathbb{C}}[x]$, thus, a basis of the vector space
$H_{F_{0}}$ can be obtained by the algorithm [16, 21]. An algorithm for
computing a basis of $H_{M_{(\gamma,f)}}$ is the following.

Algorithm 3. ( A basis of $H_{M_{(\gamma,f)}}$ )

Input: $f\in\operatorname{\mathbb{C}}[x]$: a polynomial with an isolated
singularity at $O$. $\gamma\in\operatorname{\mathbb{Q}}$: a root of
$\tilde{b}_{f,0}(s)$. Fix a monomial order $\succ$ on
$\operatorname{\mathbb{C}}[\xi]$.
Output: $\Psi$: a basis of the vector space $H_{M(\gamma,f)}$.
BEGIN
$G_{(\gamma,f)}\leftarrow$ Compute $G_{(\gamma,f)}$;
$\\{f_{1},\ldots,f_{r}\\}\leftarrow
G_{(\gamma,f)}\cap\operatorname{\mathbb{C}}[x]$;
$F_{0}\leftarrow\\{f_{1},\ldots,f_{r}\\}\cup\\{f,\frac{\partial f}{\partial
x_{1}},\ldots,\frac{\partial f}{\partial x_{n}}\\}$; $P_{1}\leftarrow
G_{(\gamma,f)}\backslash\\{f_{1},\ldots,f_{r}\\}$;
$G_{0}\leftarrow$ Compute a basis of the vector space $H_{F_{0}}$; `/*`
$G_{0}$ `is echelon form. */`
$\Psi\leftarrow\emptyset$; $L\leftarrow\emptyset$;
while $G_{0}\neq\emptyset$ do
Select $\psi$ whose head monomial is the smallest in
$\operatorname{hm}(G_{0})$; $G_{0}\leftarrow G_{0}\backslash\\{\psi\\}$;
$\displaystyle\varphi\leftarrow\psi+\sum_{\varsigma_{i}\in
L}c_{i}\varsigma_{i}$; `/*The symbol` $c_{i}$ `is an indeterminate.*/`
$E\leftarrow$ Make a system of linear equations with $c_{i}$ from
$\\{p\ast\psi=0|p\in P_{1}\\}$;
if $E$ has a solution then
$\varphi^{\prime}\leftarrow$ Substitute the solution into $c_{i}$ of
$\varphi$;
$\Psi\leftarrow\Psi\cup\\{\varphi^{\prime}\\}$;
else
$L\leftarrow L\cup\\{\psi\\}$; `/* candidate of lower cohomology classes.*/`
end-if
end-while
return $\Psi$;
END

######  Theorem 4.3.

Algorithm 3 returns a basis of the vector space $H_{M(\gamma,f)}$ and
terminates.

###### Proof.

As $H_{F_{0}}$ is the finite dimensional vector space, the set $G_{0}$ is
finite. Thus, this algorithm terminates. Since each element in the output
$\Psi$ satisfies linear partial differential equation $G_{(\gamma,f)},$ we
have $\Psi\subset H_{M_{(\gamma,f)}}.$ Since $H_{M_{(\gamma,f)}}\subset
H_{F_{0}},$ we have $\text{Span}(\Psi)=H_{M_{(\gamma,f)}}.$ Furthermore, each
element in $\Psi$ has a form $\displaystyle\psi+\sum_{\varsigma_{i}\in
L}c_{i}\varsigma_{i}$, they are linearly independent. Therefore, the algorithm
returns a basis of the vector space $H_{M_{(\gamma,f)}}.$ ∎

We illustrate Algorithm 3 with the following example

######  Example 3.

Let us consider
$f=x^{3}+yz^{2}+y^{7}+xy^{5}+xz^{2}\in\operatorname{\mathbb{C}}[x,y,z]$ that
defines an isolated singularity at the origin. By computing $\tilde{b}_{f,0}$,
we have rational numbers $\gamma=-\frac{19}{21}$ and $-\frac{4}{3}$ as roots
of $\tilde{b}_{f,0}(\gamma)=0$. Let us execute Algorithm 3 to get bases of the
vector spaces $H_{M_{(-\frac{19}{21},f)}}$ and $H_{M_{(-\frac{4}{3},f)}}$. Let
$\xi,\eta,\zeta$ denote symbols correspond to the variables $x,y,z$, and
$\partial_{x}=\frac{\partial}{\partial x}$,
$\partial_{y}=\frac{\partial}{\partial y}$,
$\partial_{z}=\frac{\partial}{\partial z}$. The monomial order $\succ$ is the
degree lexicographic with $\xi\succ\eta\succ\zeta$.

  1. $\bullet$

In case $\gamma=-\frac{19}{21}$, then $G_{(-\frac{19}{21},f)}=\\{x,y,z\\}$.
Thus, it is obvious that
$H_{M_{(-\frac{19}{21},f)}}=\text{Span}\left(1\right)$.

  2. $\bullet$

In case $\gamma=-\frac{4}{3}$, then
$G_{(-\frac{4}{3},f)}=\\{z^{2},yx,xz,xy,x^{2},12x-y^{3},24x\partial_{y}+x+2y^{2},81xz\partial_{z}-1367x+32y^{3}-1152y^{2}\partial_{y}-48y^{2}-768yz\partial_{z}-5376y,41472x\partial_{x}+61503x+1064y^{2}+13824y\partial_{y}+576y+41472z\partial_{z}+138240,35378x+124416y^{2}\partial_{x}+184509y^{2}-497664y\partial_{y}^{2}-41472y\partial_{y}+75744y-1492992z\partial_{y}\partial_{z}-62208z\partial_{z}-3981312\partial_{y}-165888\\}$.

Let $P_{0}=G_{(-\frac{4}{3},f)}\cap\operatorname{\mathbb{C}}[x,y,z]$ and

$F_{0}=P_{0}\cup\\{f,\frac{\partial f}{\partial x},\frac{\partial f}{\partial
y},\frac{\partial f}{\partial
z}\\}=\\{z^{2},zy,zx,yx,x^{2},12x-y^{3},f,\frac{\partial f}{\partial
x},\frac{\partial f}{\partial y},\frac{\partial f}{\partial z}\\}$

A basis $G_{0}$ of the vector space of $H_{F_{0}}$ is
$G_{0}=\\{1,\zeta,\eta,\eta^{2},\eta^{3}+\frac{1}{12}\xi\\}$. Set
$P_{1}=G_{(-\frac{4}{3},f)}\backslash
P_{0}=\\{24x\partial_{y}+x+2y^{2},81xz\partial_{z}-1367x+32y^{3}-1152y^{2}\partial_{y}-48y^{2}-768yz\partial_{z}-5376y,41472x\partial_{x}+61503x+1064y^{2}+13824y\partial_{y}+576y+41472z\partial_{z}+138240,35378x+124416y^{2}\partial_{x}+184509y^{2}-497664y\partial_{y}^{2}-41472y\partial_{y}+75744y-1492992z\partial_{y}\partial_{z}-62208z\partial_{z}-3981312\partial_{y}-165888\\}$,
$\Psi=\emptyset$ and $L=\emptyset$.

  3. (1)

Take $1$ whose head monomial is the smallest in $\operatorname{hm}(G_{0})$
w.r.t. $\succ$. Renew $G_{0}$ as $G_{0}\backslash\\{1\\}$ and take
$41472x\partial_{x}+61503x+1064y^{2}+13824y\partial_{y}+576y+41472z\partial_{z}+138240$
from $P_{1}$. Then, it is clear that
$(41472x\partial_{x}+61503x+1064y^{2}+13824y\partial_{y}+576y+41472z\partial_{z}+138240)\ast(1)\neq
0$. Thus, renew $L$ as $L\cup\\{1\\}$.

  4. (2)

Take $\zeta$ whose head monomial is the smallest in $\operatorname{hm}(G_{0})$
w.r.t. $\succ$. Renew $G_{0}$ as $G_{0}\backslash\\{\zeta\\}$. Set
$\varphi=\zeta+c$ where $c$ is an indeterminate. Then $\\{p\ast\varphi|p\in
P_{1}\\}=\\{0,0,41472c,-62208c+1492992c\eta\\}.$ Thus, when $c=0$, then
$\varphi\in H_{M_{(-\frac{4}{3},f)}}$. Renew $\Psi$ as $\Psi\cup\\{\zeta\\}$.

  5. (3)

Take $\eta$ whose head monomial is the smallest in $\operatorname{hm}(G_{0})$.
Renew $G_{0}$ as $G_{0}\backslash\\{\eta\\}$. Set $\varphi=\eta+c$ where $c$
is an indeterminate. Then, $\\{p\ast\varphi|p\in
P_{1}\\}=\\{0,-2304,(41472c+576)+27648\eta,(75744-62208c)+(1492992c-20736)\eta\\}.$
Obviously, the second element $-2304$ is not zero. Thus, renew $L$ as
$\\{\eta,1\\}$.

  6. (4)

Take $\eta^{2}$ whose head monomial is the smallest in
$\operatorname{hm}(G_{0})$. Renew $G_{0}$ as $G_{0}\backslash\\{\eta^{2}\\}$.
Set $\varphi=\eta^{2}+c_{1}\eta+c_{2}$ where $c_{1},c_{2}$ are indeterminates.
Then, $(24x\partial_{y}+x+2y^{2})\ast\varphi=2\neq 0$ where
$24x\partial_{y}+x+2y^{2}\in P_{1}$. Hence, $\varphi\notin
H_{M_{(-\frac{4}{3},f)}}$ and renew $L$ as $\\{\eta^{2},\eta,1\\}$.

  7. (5)

Take $\eta^{3}+\frac{1}{12}\xi$ from $G_{0}$. Renew $G_{0}$ as
$G_{0}\backslash\\{\eta^{3}+\frac{1}{12}\xi\\}$ where $c_{1},c_{2},c_{3}$ are
indeterminates. Set
$\varphi=\eta^{3}+\frac{1}{12}\xi+c_{1}\eta^{2}+c_{2}\eta+c_{3}$. Then,
$\\{p\ast\varphi|p\in
P_{1}\\}=\\{2c_{1}+\frac{1}{12},48c_{1}+2304c_{2}+\frac{266}{3}+(1152c_{1}+48)\eta,1064c_{1}+576c_{2}+41472c_{3}+\frac{20501}{4}+(576c_{1}+27648c_{2}+1064)\eta+(13824c_{1}+576)\eta^{2},184$
$509c_{1}+75744c_{2}-62208c_{3}+\frac{17689}{6}+(75744c_{1}-20736c_{2}+1492992c_{3}+184509)\eta-(124416c_{1}+5184)\eta^{2}+(20736c_{1}+1990656c_{2}+75744)\eta^{2}+(1492992c_{1}+62208)\eta^{3}\\}$
Solve the following system of linear equations that are from
$\\{p\ast\varphi|p\in P_{1}\\}.$
$2c_{1}+\frac{1}{12}=0,48c_{1}+2304c_{2}+\frac{266}{3}=0,1152c_{1}+48=0,1064c_{1}+576c_{2}+41472c_{3}+\frac{20501}{4}=0,576c_{1}+27648c_{2}+1064=0,13824c_{1}+576=0,184509c_{1}+75744c_{2}-62208c_{3}+\frac{17689}{6}=0,75744c_{1}-20736c_{2}+1492992c_{3}+184509=0,124416c_{1}+5184=0,20736c_{1}+1990656c_{2}+75744=0,1492992c_{1}+62208=0.$
Then, $c_{1}=-\frac{1}{24},c_{2}=-\frac{65}{1728},c_{3}=0$. Renew $\Psi$ as
$\\{\eta,\eta^{3}+\frac{1}{12}\xi-\frac{1}{24}\eta^{2}-\frac{65}{1728}\eta\\}$.

  8. (6)

Since $G_{0}=\emptyset$, Algorithm 3 stops. Therefore, $\Psi$ is a basis of
the vector space $H_{M_{(-\frac{4}{3},f)}}$.

By using a framework presented in [16], we have extended Algorithm 3 to handle
parametric cases. The resulting algorithm that compute the parametric local
cohomology solution space of parametric holonomic $D$-module $M_{(\gamma,f)}$
is implemented in the computer algebra system Risa/Asir.

##  5\. New algorithm

Here we introduce a new algorithm for computing reduced $b$-functions
$\tilde{b}_{f,0}$ of a semi-weighted homogeneous polynomial. As we described
in subsection 3.2, the computational complexity of computing a Gröbner basis
of $\text{Ann}(f^{s})+\operatorname{{\it Id}}(f,\frac{\partial f}{\partial
x_{1}},\ldots,\frac{\partial f}{\partial x_{n}})$ is quite high. In order to
overcome the difficulty, we adopt the idea introduced by Levandovskyy and
Martin-Morales [12] and address to the computation of $b$-functions associated
with $\mu$-constant deformations.

Let $f=f_{0}+g\in\operatorname{\mathbb{C}}[x]$ where $f_{0}$ is the weighted
homogeneous part, $d\in{\mathbb{N}}$, ${\bf
w}=(w_{1},\ldots,w_{n})\in\operatorname{\mathbb{N}}^{n}$ and $g$ is a linear
combination of upper monomials.

###  5.1. Properties of semi-weighted homogeneous singularities

First we review some properties of semi-weighted homogeneous singularities
that are needed for constructing a new algorithm.

######  Definition 5.1.

Let $f_{0}$ be a weighted homogeneous polynomial of type $(d;{\bf w})$ with an
isolated singularity at the origin $O.$ The Poincaré polynomial of $f_{0}$ is
the univariate polynomial defined to be

$P_{(d;{\bf
w})}(t)=\frac{t^{d-w_{1}}-1}{t^{w_{1}}-1}\cdot\frac{t^{d-w_{2}}-1}{t^{w_{2}}-1}\cdots\frac{t^{d-w_{n}}-1}{t^{w_{n}}-1}.$

It is well-known that all roots of $\tilde{b}_{f_{0},0}$ can be computed by
the Poincaré polynomial.

######  Theorem 5.2.

Let $f_{0}$ be a weighted homogeneous polynomial of type $(d;{\bf w})$ with an
isolated singularity at the origin $O$, and $w_{0}=\sum_{i=1}^{n}w_{i}$. Let
$P_{(d;{\bf w})}(t)=\sum_{i=1}^{r}c_{i}t^{\alpha_{i}}$ ($c_{i}\neq 0$) be the
Poincaré polynomial of type $(d;{\bf w})$. Then, the set of roots of
$\tilde{b}_{f_{0},0}(s)=0$ is equal to
$\left\\{{-\frac{\alpha_{i}+w_{0}}{d}}\left|\,\vphantom{-\frac{\alpha_{i}+w_{0}}{d}}{1\leq
i\leq r}\right.\right\\}$.

The following properties are from [6, 22].

######  Theorem 5.3.

Let $\\{\gamma_{1},\ldots,\gamma_{r}\\}$ be the set of roots of
$\tilde{b}_{f,0}(s)=0$ and let $\mu$ be the Milnor number of the singularity
$f=0$ at the origin $O$. Then,
$\displaystyle\mu=\sum_{i=1}^{r}\dim_{\operatorname{\mathbb{C}}}(H_{M_{(\gamma_{i},f)}}).$

###  5.2. New algorithm

The new algorithm mainly consists of the following three steps,

  1. Step 1:

to compute candidates $\gamma$s of the roots of $\tilde{b}_{f,0}(s)=0$,

  2. Step 2:

to check whether $\tilde{b}_{f,0}(\gamma)=0$ or $\tilde{b}_{f,0}(\gamma)\neq
0$,

  3. Step 3:

to check whether $\Gamma=\\{\gamma_{1},\ldots,\gamma_{r}\\}$ is equal to
$\\{\gamma|\tilde{b}_{f,0}(\gamma)=0\\}$ or not, where
$\gamma_{1},\ldots,\gamma_{r}$ are roots of $\tilde{b}_{f,0}(s)=0$.

The following lemma quoted from [7], tells us how to compute the candidates.

######  Lemma 5.4.

Let $E_{0}=\\{\gamma\in\mathbb{Q}|\tilde{b}_{f_{0},0}(\gamma)=0\\}$ and
$E=\\{\gamma\in\mathbb{Q}|\tilde{b}_{f,0}(\gamma)=0\\}$. Then, $E$ is a subset
of $E^{\prime}=\\{\gamma+k|\gamma\in E_{0},k\in\mathbb{Z},-n<\gamma+k<0\\}$
where $\mathbb{Z}$ is the set of integers.

Since $E_{0}$ is determined by Theorem 5.2, it is easy to obtain $E^{\prime}.$
Empirically, it is sufficient to check $k=0,1,2$. Hence, in Step 1, we use

$E^{\prime}=\\{\gamma+k|\gamma\in E_{0},k\in\\{0,1,2\\},-n<\gamma+k<0\\}$

as s set of candidates of the roots.

Next, in Step 2, we have to check whether $\gamma\in E^{\prime}$ is a root of
$\tilde{b}_{f,0}(s)=0$ or not. We borrow the idea of Levandovskyy and Martin-
Morales [12].

######  Lemma 5.5.

Let $H$ be a basis of $\text{Ann}(f^{s})$ in $D[s]$ and
$\gamma\in\operatorname{\mathbb{Q}}$. Let $G$ be a minimal Gröbner basis of
$\operatorname{{\it Id}}(H\cup\\{f,\frac{\partial f}{\partial
x_{1}},\frac{\partial f}{\partial x_{2}},\ldots,$ $\frac{\partial f}{\partial
x_{n}}\\}\cup\\{s-\gamma\\})$ w.r.t. a block order with $x\cup\partial\gg s$.
Then, if $s-\gamma\in G$, $s-\gamma$ is a factor of the global $b$-function of
$f$.

Remark: The computational speed of computing a minimal Gröbner basis of
$\operatorname{{\it Id}}(H\cup\\{f,\frac{\partial f}{\partial x_{1}},\ldots,$
$\frac{\partial f}{\partial x_{n}}\\}\cup\\{s-\gamma\\})$ is much faster than
that of $\operatorname{{\it Id}}(H\cup\\{f,\frac{\partial f}{\partial
x_{1}},\ldots,\frac{\partial f}{\partial x_{n}}\\})$, because the degree of
$s-\gamma$ is 1. Moreover, for a candidate $\gamma\in E^{\prime}$, it is
sufficient to check the single root because of Theorem 3.4. However, as we are
considering “local” $b$-functions at the origin $O$, we need to check the
support of $\gamma$ (i.e., $\text{supp}(M_{(\gamma,f)})$) if $s-\gamma\in G$.

As we know how to compute a CGS in $D[s]$, we can naturally extend the idea to
parametric cases. We have implemented the parametric version of Lemma 5.5 in
Risa/Asir.

In our implementation, the command `para_ann1` (or `para_ann`) returns a CGS
of $\text{Ann}(f^{s})$, and the command `root_check` (or `root_check11`)
returns a CGS of $\operatorname{{\it
Id}}(\text{Ann}(f^{s})\cup\\{f,\frac{\partial f}{\partial
x_{1}},\frac{\partial f}{\partial x_{2}},\ldots,\frac{\partial f}{\partial
x_{n}}\\}\cup\\{s-\gamma\\})$ where $\gamma\in\operatorname{\mathbb{Q}}$. In
Example 4, `ANN` is a CGS of $\text{Ann}(f^{s})$. The form `[[S1],[S2]]` means
a stratum
$\operatorname{\mathbb{V}}($`S1`$)\backslash\operatorname{\mathbb{V}}($`S2`$)$.

######  Example 4.

Let us consider $S_{16}$ singularity. The $\mu$-constant deformation is given
by $f=x^{2}z+yz^{2}+xy^{4}+u_{1}y^{6}+u_{2}z^{3}$ where $u_{1},u_{2}$ are
parameters. Let us check whether $17s+19$ is a factor of $\tilde{b}_{f}(s)$ or
not.

    [2727] F=x^2*z+y*z^2+x*y^4+u1*y^6+u2*z^3$
    [2728] ANN=para_ann1(F,[u1,u2],[x,y,z])$
    [2729] roots_check(ANN,F,17*s+19,[u1,u2],[x,y,z]);
    [[0],[486*u2*u1^13-1143*u2^2*u1^9-639*u2^3*u1^5-68*u2^4*u1]]
    [17*s+19,z,x,y^2,(4030*u1^4+4913*dy*u1-9826*u2)*y+11560*u2^2*z+9826*u1]

    [[27*u1^4+4*u2],[u1,u2]]
    [17*s+19,z,x,-y^2,(140711*u1^3+9826*dy)*y+19652]

    [[u1,u2],[1]]
    [1]

    [[6*u1^4-17*u2],[u1,u2]]
    [17*s+19,z,x,y^2,(-562*u1^3-4913*dy)*y-9826]

    [[3*u1^4+u2],[u1,u2]]
    [17*s+19,z,x,y^2,(33508*u1^3+4913*dy)*y+9826]

    [[u1],[u1,u2]]
    [17*s+19,z,y,x]

    [[u2],[u1,u2]]
    [17*s+19,z,x,y^2,(4030*u1^3+4913*dy)*y+9826]

The monomial order $\succ$ used in the computation above is a block order
$\\{\partial_{x},\partial_{y},\partial_{z}\\}\gg\\{x,y,z\\}\gg s$ which is
specified on $\text{Mono}(\\{\partial_{x},\partial_{y},\partial_{z}\\})$ as
the total degree lexicographic monomial order with
$\partial_{x}\succ\partial_{y}\succ\partial_{z}$, and on
$\operatorname{Mono}(\\{x,y,z\\})$ as the total degree lexicographic monomial
order with $z\succ y\succ x$.

Let $I$ denote the ideal generated by $\text{Ann}(f^{s})$ and
$\\{f,\frac{\partial f}{\partial x},\frac{\partial f}{\partial
y},\frac{\partial f}{\partial z},17s+19\\}$ in the ring $(D[u_{1},u_{2}])[s]$,
i.e., $=\operatorname{{\it Id}}(\text{Ann}(f^{s})\cup\\{f,\frac{\partial
f}{\partial x},\frac{\partial f}{\partial y},\frac{\partial f}{\partial
z},17s+19\\})$.

The meanings of the output above are the following.

  1. (1)

If $(u_{1},u_{2})$ belongs to
$\operatorname{\mathbb{U}}_{1}=\left(\operatorname{\mathbb{C}}^{2}\backslash\operatorname{\mathbb{V}}(486u_{2}u_{1}^{13}-1143u_{2}^{2}u_{1}^{9}-639u_{2}^{3}u_{1}^{5}-68u_{2}^{4}u_{1})\right)=\left(\operatorname{\mathbb{C}}^{2}\backslash\operatorname{\mathbb{V}}((27u_{1}^{4}+\right.$
$\left.4u_{2})(6u_{1}^{4}-17u_{2})(3u_{1}^{4}+u_{2})u_{1}u_{2})\right)$, then

$G_{1}=\\{17s+19,z,x,y^{2},(4030u_{1}^{4}+4913\partial_{y}u_{1}-9826u_{2})y+11560u_{2}^{2}z+9826u_{1}\\}$

is a minimal Gröbner basis of $I$ w.r.t. $\succ$. As $17s+19\in G_{1}$ and
$G_{1}\cap\mathbb{C}[x,y,z]=\\{z,x,y^{2}\\}$, $17s+19$ is a factor of
$\tilde{b}_{f}(s)$ and $\text{supp}(M_{(-\frac{19}{17},f)})=\\{O\\}$, namely,
$17s+19$ is a factor of $\tilde{b}_{f,0}(s)$.

  2. (2)

If $(u_{1},u_{2})$ belongs to
$\operatorname{\mathbb{U}}_{2}=\operatorname{\mathbb{V}}(27u_{1}^{4}+4u_{2})\backslash\operatorname{\mathbb{V}}(u_{1},u_{2})$,
then $G_{2}=\\{17s+19,z,x,-y^{2},(140711$
$u_{1}^{3}+9826\partial_{y})y+19652\\}$ is a minimal Gröbner basis of $I$
w.r.t. $\succ$. As $17s+19\in G_{2}$ and
$G_{2}\cap\operatorname{\mathbb{C}}[x,y,z]=\\{z,x,-y^{2}\\}$, $17s+19$ is a
factor of $\tilde{b}_{f,0}(s)$.

  3. (3)

If $(u_{1},u_{2})$ belongs to
$\operatorname{\mathbb{U}}_{3}=\operatorname{\mathbb{V}}(u_{1},u_{2})$, then
$G_{3}=\\{1\\}$ is a minimal Gröbner basis of $I$ w.r.t. $\succ$. As
$17s+19\notin G_{3}$, $17s+19$ is not a factor of $\tilde{b}_{f}(s)$.

  4. (4)

If $(u_{1},u_{2})$ belongs to
$\operatorname{\mathbb{U}}_{4}=\operatorname{\mathbb{V}}(6u_{1}^{4}-17u_{2})\backslash\operatorname{\mathbb{V}}(u_{1},u_{2})$,
then $G_{4}=\\{17s+19,z,x,y^{2},(-562u_{1}^{3}-4913\partial_{y})y-9826\\}$ is
a minimal Gröbner basis of $I$ w.r.t. $\succ$. As $17s+19\in G_{4}$ and
$G_{4}\cap\operatorname{\mathbb{C}}[x,y,z]=\\{z,x,y^{2}\\}$, $17s+19$ is a
factor of $\tilde{b}_{f,0}(s)$.

  5. (5)

If $(u_{1},u_{2})$ belongs to
$\operatorname{\mathbb{U}}_{5}=\operatorname{\mathbb{V}}(3u_{1}^{4}+u_{2})\backslash\operatorname{\mathbb{V}}(u_{1},u_{2})$,
then $G_{5}=\\{17s+19,z,x,y^{2},(33508u_{1}^{3}+4913\partial_{y})y+9826\\}$ is
a minimal Gröbner basis of $I$ w.r.t. $\succ$. As $17s+19\in G_{5}$ and
$G_{5}\cap\operatorname{\mathbb{C}}[x,y,z]=\\{z,x,y^{2}\\}$, $17s+19$ is a
factor of $\tilde{b}_{f,0}(s)$.

  6. (6)

If $(u_{1},u_{2})$ belongs to
$\operatorname{\mathbb{U}}_{6}=\operatorname{\mathbb{V}}(u_{1})\backslash\operatorname{\mathbb{V}}(u_{1},u_{2})$,
then $G_{6}=\\{17s+19,z,y,x\\}$ is a minimal Gröbner basis of $I$ w.r.t.
$\succ$. As $17s+19\in G_{6}$ and
$G_{6}\cap\operatorname{\mathbb{C}}[x,y,z]=\\{z,y,x\\}$, $17s+19$ is a factor
of $\tilde{b}_{f,0}(s)$.

  7. (7)

If $(u_{1},u_{2})$ belongs to
$\operatorname{\mathbb{U}}_{7}=\operatorname{\mathbb{V}}(u_{2})\backslash\operatorname{\mathbb{V}}(u_{1},u_{2})$,
then $G_{7}=\\{17s+19,z,x,y^{2},(4030u_{1}^{3}+4913\partial_{y})y+9826\\}$ is
a minimal Gröbner basis of $I$ w.r.t. $\succ$. As $17s+19\in G_{7}$ and
$G_{7}\cap\operatorname{\mathbb{C}}[x,y,z]=\\{z,x,y^{2}\\}$, $17s+19$ is a
factor of $\tilde{b}_{f,0}(s)$.

Note that $17s+19\in G_{i}\ (i=1,2,4,5,6,7)$ and $17s+19\notin G_{3}$.

Since

$\operatorname{\mathbb{U}}_{1}\cup\operatorname{\mathbb{U}}_{2}\cup\operatorname{\mathbb{U}}_{4}\cup\operatorname{\mathbb{U}}_{5}\cup\operatorname{\mathbb{U}}_{6}\cup\operatorname{\mathbb{U}}_{7}=\operatorname{\mathbb{C}}^{2}\backslash\operatorname{\mathbb{U}}_{3},$

we have
i) if $(u_{1},u_{2})\neq(0,0)$, then $17s+19$ is a factor of
$\tilde{b}_{f}(s)$ and $\text{supp}(M_{(-\frac{19}{17},f)})=\\{O\\}$, namely,
$17s+19$ is a factor of (local) $\tilde{b}_{f,0}(s)$,
ii) if $u_{1}=u_{2}=0$, then $17s+19$ is not a factor of (global)
$\tilde{b}_{f}(s)$.

In Step 3, we apply Theorem 5.3 and we use Algorithm 3 for computing ${\rm
dim}_{{\mathbb{C}}}(H_{M_{(\gamma,f)}}).$

Algorithm 4. (Local $b$-function at $O$)

Input: $f=f_{0}+g\in\operatorname{\mathbb{C}}[x]$: a semi-weighted homogeneous
polynomial with an isolated singularity at $O$.
$H$: a basis of $\text{Ann}(f^{s})$. $\mu(f_{0})\in\operatorname{\mathbb{N}}$:
the Milnor number of $f_{0}$ at the origin $O$.
$\succ$ : a monomial order satisfying $\\{x,\partial\\}\gg s$.
Output: $b$: the reduced $b$-function $\tilde{b}_{f,0}(s)$.
BEGIN
$b\leftarrow 1$; $\mu\leftarrow 0$; $k\leftarrow 0$;
$J\leftarrow\\{f,\frac{\partial f}{\partial x_{1}},\ldots,\frac{\partial
f}{\partial x_{n}}\\}$;
$E_{0}\leftarrow$ Compute all roots of $\tilde{b}_{f_{0},0}(s)$ by the
Poincaré polynomial;
while $\mu\neq\mu(f_{0})$ do
$E^{\prime}\leftarrow\\{\gamma+k|\gamma\in E_{0},-n<\gamma+k<0\\}$;
while $E^{\prime}\neq\emptyset$ do
Select $\gamma$ from $E^{\prime}$; $E^{\prime}\leftarrow
E^{\prime}\backslash\\{\gamma\\}$;
$G\leftarrow$ Compute a minimal Gröbner basis of $\operatorname{{\it
Id}}(H\cup J\cup\\{s-\gamma\\})$ w.r.t. $\succ$;
if $\left(s-\gamma\in G\right)$ and
$\left(O\in\text{supp}(M_{(\gamma,f)})\right)$ then
$b\leftarrow b\cdot(s-\gamma)$;
$\mu\leftarrow\mu+\dim_{\operatorname{\mathbb{C}}}(H_{M_{(\gamma,f)}})$;
end-if
end-while
$k\leftarrow k+1$;
end-while
return $b$;
END

As we described in section 3 and 4, a CGS of $\text{Ann}(f^{s})$ and a basis
of the vector space $H_{M_{(\gamma,f)}}$ with parameters, are computable.
Thus, Algorithm 4 can be generalized to parametric cases, too. We have
computed all $b$-functions of Table 1 by utilizing the generalized algorithm.
We have obtained all $b$-functions of Table 1, successfully. The $b$-functions
of Table 1 are given in Section 6.

Remark: We tried to compute local $b$-functions associated with $\mu$-constant
deformations of inner modality 3 singularities by using a computer PC1 (see
section 3). We could obtain none of CGSs of $\text{Ann}(f^{s})$ within three
months and thus we could not use the method described in section 5. We expect
however that local $b$-functions associated with $\mu$-constant deformations
can be computed by using the proposed method provided that parametric bases of
$\text{Ann}(f^{s})$ is given.

In this paper, we introduce a new algorithm for computing parametric
$\tilde{b}_{f,0}$ of a semi-weighted homogeneous polynomial by improving
Algorithm 2. The resulting algorithm has better performance than Algorithm 2.
The resulting algorithm and Algorithm 2 use the same CGS algorithm described
in [14, 15]. The difference lies in the way of its using. The versatility of
CGS algorithm allows the specialization and the improvement.

##  6\. List of $b$-functions associated with $\mu$-constant deformations

Here all $b$-functions $\tilde{b}_{f,0}(s)$ of $\mu$-constant deformation of
inner modality 2 singularities, are presented.

Currently, our Risa/Asir implementation is in the following webpage

https://www-math.ias.tokushima-u.ac.jp/~nabesima/bfunction/bfunc2.html.

Two computation times “CGS of $\text{Ann}(f^{s})$” and “parametric version of
Algorithm 4” are also presented in each $\mu$-constant deformation. The time
is given in CPU seconds. The computer [OS: Windows 10, CPU: intel core
i9-7900X, 3.30 GHz, Memory: 128 GB] was used.

  1. $\bullet$ $E_{18}$:

$f=x^{3}+y^{10}+u_{1}xy^{7}+u_{2}xy^{8}$

stratum | $\tilde{b}_{f,0}(s)$
---|---
$\operatorname{\mathbb{C}}^{2}\backslash\operatorname{\mathbb{V}}(u_{1})$ | $b_{\text{st}}(s)(s+\frac{14}{30})(s+\frac{17}{30})$
$\operatorname{\mathbb{V}}(u_{1})\backslash\operatorname{\mathbb{V}}(u_{1},u_{2})$ | $b_{\text{st}}(s)(s+\frac{17}{30})(s+\frac{44}{30})$
$\operatorname{\mathbb{V}}(u_{1},u_{2})$ | $b_{\text{st}}(s)(s+\frac{44}{30})(s+\frac{47}{30})$

$b_{\text{st}}(s)=(s+\frac{13}{30})(s+\frac{16}{30})(s+\frac{19}{30})(s+\frac{22}{30})(s+\frac{23}{30})(s+\frac{25}{30})(s+\frac{26}{30})$

$\times(s+\frac{28}{30})(s+\frac{29}{30})(s+\frac{31}{30})(s+\frac{32}{30})(s+\frac{34}{30})(s+\frac{35}{30})(s+\frac{37}{30})(s+\frac{38}{30})(s+\frac{41}{30}).$

CGS of $\text{Ann}(f^{s})$: 3141 |  | parametric version of Algorithm 4: 5.156 |
---|---|---|---

  2. $\bullet$ $E_{19}$:

$f=x^{3}+xy^{7}+u_{1}y^{11}+u_{2}y^{12}$

$\operatorname{\mathbb{C}}^{2}\backslash\operatorname{\mathbb{V}}(u_{1})$ | $b_{\text{st}}(s)(s+\frac{10}{21})(s+\frac{12}{21})$
---|---
$\operatorname{\mathbb{V}}(u_{1})\backslash\operatorname{\mathbb{V}}(u_{1},u_{2})$ | $b_{\text{st}}(s)(s+\frac{12}{21})(s+\frac{31}{21})$
$\operatorname{\mathbb{V}}(u_{1},u_{2})$ | $b_{\text{st}}(s)(s+\frac{31}{21})(s+\frac{33}{21})$

$b_{\text{st}}(s)=(s+\frac{9}{21})(s+\frac{11}{21})(s+\frac{13}{21})(s+\frac{15}{21})(s+\frac{16}{21})(s+\frac{17}{21})(s+\frac{18}{26})(s+\frac{19}{21})$

$\times(s+\frac{20}{21})(s+\frac{21}{21})(s+\frac{22}{21})(s+\frac{23}{21})(s+\frac{24}{21})(s+\frac{25}{21})(s+\frac{26}{21})(s+\frac{27}{21})(s+\frac{29}{21}).$

CGS of $\text{Ann}(f^{s})$: 10410 |  | parametric version of Algorithm 4: 45.5 |
---|---|---|---

  3. $\bullet$ $E_{20}$:

$f=x^{3}+y^{11}+u_{1}xy^{8}+u_{2}xy^{9}$

$\operatorname{\mathbb{C}}^{2}\backslash\operatorname{\mathbb{V}}(u_{1})$ | $b_{\text{st}}(s)(s+\frac{16}{33})(s+\frac{19}{33})$
---|---
$\operatorname{\mathbb{V}}(u_{1})\backslash\operatorname{\mathbb{V}}(u_{1},u_{2})$ | $b_{\text{st}}(s)(s+\frac{19}{33})(s+\frac{49}{33})$
$\operatorname{\mathbb{V}}(u_{1},u_{2})$ | $b_{\text{st}}(s)(s+\frac{49}{33})(s+\frac{52}{33})$

$b_{\text{st}}(s)=(s+\frac{14}{33})(s+\frac{17}{33})(s+\frac{20}{33})(s+\frac{23}{33})(s+\frac{25}{33})(s+\frac{26}{33})(s+\frac{28}{33})(s+\frac{29}{33})(s+\frac{31}{33})$

$\times(s+\frac{32}{33})(s+\frac{34}{33})(s+\frac{35}{33})(s+\frac{37}{33})(s+\frac{38}{33})(s+\frac{40}{33})(s+\frac{41}{33})(s+\frac{43}{33})(s+\frac{46}{33}).$

CGS of $\text{Ann}(f^{s})$: 2667 |  | parametric version of Algorithm 4: 164 |
---|---|---|---

  4. $\bullet$ $W_{17}$:

$f=x^{4}+xy^{5}+u_{1}y^{7}+u_{2}y^{8}$

$\operatorname{\mathbb{C}}^{2}\backslash\operatorname{\mathbb{V}}(u_{1})$ | $b_{\text{st}}(s)(s+\frac{9}{20})(s+\frac{12}{20})$
---|---
$\operatorname{\mathbb{V}}(u_{1})\backslash\operatorname{\mathbb{V}}(u_{1},u_{2})$ | $b_{\text{st}}(s)(s+\frac{12}{20})(s+\frac{29}{20})$
$\operatorname{\mathbb{V}}(u_{1},u_{2})$ | $b_{\text{st}}(s)(s+\frac{29}{20})(s+\frac{32}{20})$

$b_{\text{st}}(s)=(s+\frac{8}{20})(s+\frac{11}{20})(s+\frac{13}{20})(s+\frac{14}{20})(s+\frac{16}{20})(s+\frac{17}{20})(s+\frac{18}{20})(s+\frac{19}{20})$

$\times(s+\frac{20}{20})(s+\frac{21}{20})(s+\frac{22}{20})(s+\frac{23}{20})(s+\frac{24}{20})(s+\frac{26}{20})(s+\frac{27}{20}).$

CGS of $\text{Ann}(f^{s})$: 354.1 |  | parametric version of Algorithm 4: 7.547 |
---|---|---|---

  5. $\bullet$ $W_{18}$:

$f=x^{4}+y^{7}+u_{1}x^{2}y^{4}+u_{2}x^{2}y^{5}$

$\operatorname{\mathbb{C}}^{2}\backslash\operatorname{\mathbb{V}}(u_{1})$ | $b_{\text{st}}(s)(s+\frac{13}{28})(s+\frac{17}{28})$
---|---
$\operatorname{\mathbb{V}}(u_{1})\backslash\operatorname{\mathbb{V}}(u_{1},u_{2})$ | $b_{\text{st}}(s)(s+\frac{17}{28})(s+\frac{41}{28})$
$\operatorname{\mathbb{V}}(u_{1},u_{2})$ | $b_{\text{st}}(s)(s+\frac{41}{28})(s+\frac{45}{28})$

$b_{\text{st}}(s)=(s+\frac{11}{28})(s+\frac{15}{28})(s+\frac{18}{28})(s+\frac{19}{28})(s+\frac{22}{28})(s+\frac{23}{28})(s+\frac{25}{28})(s+\frac{26}{28})$

$\times(s+\frac{27}{28})(s+\frac{29}{28})(s+\frac{30}{28})(s+\frac{31}{28})(s+\frac{33}{28})(s+\frac{34}{28})(s+\frac{37}{28})(s+\frac{38}{28}).$

CGS of $\text{Ann}(f^{s})$: 149.8 |  | parametric version of Algorithm 4: 2.438 |
---|---|---|---

  6. $\bullet$ $Z_{17}$:

$f=x^{3}y+y^{8}+u_{1}xy^{6}+u_{2}xy^{7}$

$\operatorname{\mathbb{C}}^{2}\backslash\operatorname{\mathbb{V}}(u_{1})$ | $b_{\text{st}}(s)(s+\frac{11}{24})(s+\frac{14}{24})$
---|---
$\operatorname{\mathbb{V}}(u_{1})\backslash\operatorname{\mathbb{V}}(u_{1},u_{2})$ | $b_{\text{st}}(s)(s+\frac{14}{24})(s+\frac{35}{24})$
$\operatorname{\mathbb{V}}(u_{1},u_{2})$ | $b_{\text{st}}(s)(s+\frac{35}{24})(s+\frac{38}{24})$

$b_{\text{st}}(s)=(s+\frac{10}{24})(s+\frac{13}{24})(s+\frac{16}{24})(s+\frac{17}{24})(s+\frac{19}{24})(s+\frac{20}{24})(s+\frac{22}{24})(s+\frac{23}{24})$

$\times(s+\frac{24}{24})(s+\frac{25}{24})(s+\frac{26}{24})(s+\frac{28}{24})(s+\frac{29}{24})(s+\frac{31}{24})(s+\frac{32}{24}).$

CGS of $\text{Ann}(f^{s})$: 903.8 |  | parametric version of Algorithm 4: 15.25 |
---|---|---|---

  7. $\bullet$ $Z_{18}$ :

$f=x^{3}y+xy^{6}+u_{1}y^{9}+u_{2}y^{10}$

$\operatorname{\mathbb{C}}^{2}\backslash\operatorname{\mathbb{V}}(u_{1})$ | $b_{\text{st}}(s)(s+\frac{8}{17})(s+\frac{10}{17})$
---|---
$\operatorname{\mathbb{V}}(u_{1})\backslash\operatorname{\mathbb{V}}(u_{1},u_{2})$ | $b_{\text{st}}(s)(s+\frac{10}{17})(s+\frac{25}{17})$
$\operatorname{\mathbb{V}}(u_{1},u_{2})$ | $b_{\text{st}}(s)(s+\frac{25}{17})(s+\frac{27}{17})$

$b_{\text{st}}(s)=(s+\frac{7}{17})(s+\frac{9}{17})(s+\frac{11}{17})(s+\frac{12}{17})(s+\frac{13}{17})(s+\frac{14}{17})(s+\frac{15}{17})(s+\frac{16}{17})$

$\times(s+\frac{17}{17})(s+\frac{18}{17})(s+\frac{19}{17})(s+\frac{20}{17})(s+\frac{21}{17})(s+\frac{22}{17})(s+\frac{23}{17}).$

CGS of $\text{Ann}(f^{s})$: 546.1 |  | parametric version of Algorithm 4: 5.344 |
---|---|---|---

  8. $\bullet$ $Z_{19}$:

$f=x^{3}y+y^{9}+u_{1}xy^{7}+u_{2}xy^{8}$

$\operatorname{\mathbb{C}}^{2}\backslash\operatorname{\mathbb{V}}(u_{1})$ | $b_{\text{st}}(s)(s+\frac{13}{27})(s+\frac{16}{27})$
---|---
$\operatorname{\mathbb{V}}(u_{1})\backslash\operatorname{\mathbb{V}}(u_{1},u_{2})$ | $b_{\text{st}}(s)(s+\frac{16}{27})(s+\frac{40}{27})$
$\operatorname{\mathbb{V}}(u_{1},u_{2})$ | $b_{\text{st}}(s)(s+\frac{40}{27})(s+\frac{43}{27})$

$b_{\text{st}}(s)=(s+\frac{11}{27})(s+\frac{14}{27})(s+\frac{17}{27})(s+\frac{19}{27})(s+\frac{20}{27})(s+\frac{22}{27})(s+\frac{23}{27})(s+\frac{25}{27})$

$\times(s+\frac{26}{27})(s+\frac{27}{27})(s+\frac{28}{27})(s+\frac{29}{27})(s+\frac{31}{27})(s+\frac{32}{27})(s+\frac{34}{27})(s+\frac{35}{27})(s+\frac{37}{27}).$

CGS of $\text{Ann}(f^{s})$: 984.2 |  | parametric version of Algorithm 4: 5.141 |
---|---|---|---

  9. $\bullet$ $Q_{16}$:

$f=x^{3}+yz^{2}+y^{7}+u_{1}xy^{5}+u_{2}xz^{2}$

$\operatorname{\mathbb{C}}^{2}\backslash\operatorname{\mathbb{V}}(u_{1})$ | $b_{\text{st}}(s)(s+\frac{20}{21})(s+\frac{23}{21})$
---|---
$\operatorname{\mathbb{V}}(u_{1})\backslash\operatorname{\mathbb{V}}(u_{1},u_{2})$ | $b_{\text{st}}(s)(s+\frac{23}{21})(s+\frac{41}{21})$
$\operatorname{\mathbb{V}}(u_{1},u_{2})$ | $b_{\text{st}}(s)(s+\frac{41}{21})(s+\frac{44}{21})$

$b_{\text{st}}(s)=(s+\frac{19}{21})(s+\frac{22}{21})(s+\frac{25}{21})(s+\frac{26}{21})(s+\frac{28}{21})(s+\frac{29}{21})(s+\frac{31}{21})(s+\frac{32}{21})$

$\times(s+\frac{34}{21})(s+\frac{35}{21})(s+\frac{37}{21})(s+\frac{38}{21}).$

CGS of $\text{Ann}(f^{s})$: 50730 |  | parametric version of Algorithm 4: 3.953 |
---|---|---|---

  10. $\bullet$ $Q_{17}$:

$f=x^{3}+yz^{2}+xy^{5}+u_{1}y^{8}+u_{2}y^{9}$

$\operatorname{\mathbb{C}}^{2}\backslash\operatorname{\mathbb{V}}(u_{1})$ | $b_{\text{st}}(s)(s+\frac{29}{30})(s+\frac{33}{30})$
---|---
$\operatorname{\mathbb{V}}(u_{1})\backslash\operatorname{\mathbb{V}}(u_{1},u_{2})$ | $b_{\text{st}}(s)(s+\frac{33}{30})(s+\frac{59}{30})$
$\operatorname{\mathbb{V}}(u_{1},u_{2})$ | $b_{\text{st}}(s)(s+\frac{59}{30})(s+\frac{63}{30})$

$b_{\text{st}}(s)=(s+\frac{27}{30})(s+\frac{31}{30})(s+\frac{35}{30})(s+\frac{37}{30})(s+\frac{39}{30})(s+\frac{40}{30})(s+\frac{41}{30})(s+\frac{43}{30})$

$\times(s+\frac{45}{30})(s+\frac{47}{30})(s+\frac{49}{30})(s+\frac{50}{30})(s+\frac{51}{30})(s+\frac{53}{30})(s+\frac{55}{30}).$

CGS of $\text{Ann}(f^{s})$: 1073 |  | parametric version of Algorithm 4: 2.813 |
---|---|---|---

  11. $\bullet$ $Q_{18}$ :

$f=x^{3}+yz^{2}+y^{8}+u_{1}xy^{6}+u_{2}xz^{2}$

$\operatorname{\mathbb{C}}^{2}\backslash\operatorname{\mathbb{V}}(u_{1})$ | $b_{\text{st}}(s)(s+\frac{47}{48})(s+\frac{53}{48})$
---|---
$\operatorname{\mathbb{V}}(u_{1})\backslash\operatorname{\mathbb{V}}(u_{1},u_{2})$ | $b_{\text{st}}(s)(s+\frac{53}{48})(s+\frac{95}{48})$
$\operatorname{\mathbb{V}}(u_{1},u_{2})$ | $b_{\text{st}}(s)(s+\frac{95}{48})(s+\frac{101}{48})$

$b_{\text{st}}(s)=(s+\frac{43}{48})(s+\frac{49}{48})(s+\frac{55}{48})(s+\frac{59}{48})(s+\frac{61}{48})(s+\frac{64}{48})(s+\frac{65}{48})(s+\frac{67}{48})$

$\times(s+\frac{71}{48})(s+\frac{73}{48})(s+\frac{77}{48})(s+\frac{79}{48})(s+\frac{80}{48})(s+\frac{83}{48})(s+\frac{85}{48})(s+\frac{89}{48}).$

CGS of $\text{Ann}(f^{s})$: 1516 |  | parametric version of Algorithm 4: 6.844 |
---|---|---|---

  12. $\bullet$ $S_{16}$ :

$f=x^{2}z+yz^{2}+xy^{4}+u_{1}y^{6}+u_{2}z^{3}$

$\operatorname{\mathbb{C}}^{2}\backslash\operatorname{\mathbb{V}}(u_{1})$ | $b_{\text{st}}(s)(s+\frac{16}{17})(s+\frac{19}{17})$
---|---
$\operatorname{\mathbb{V}}(u_{1})\backslash\operatorname{\mathbb{V}}(u_{1},u_{2})$ | $b_{\text{st}}(s)(s+\frac{19}{17})(s+\frac{33}{17})$
$\operatorname{\mathbb{V}}(u_{1},u_{2})$ | $b_{\text{st}}(s)(s+\frac{33}{17})(s+\frac{36}{17})$

$b_{\text{st}}(s)=(s+\frac{15}{17})(s+\frac{18}{17})(s+\frac{20}{17})(s+\frac{21}{17})(s+\frac{22}{17})(s+\frac{23}{17})(s+\frac{24}{17})$

$\times(s+\frac{25}{17})(s+\frac{26}{17})(s+\frac{27}{17})(s+\frac{28}{17})(s+\frac{29}{17})(s+\frac{30}{17})(s+\frac{31}{17}).$

CGS of $\text{Ann}(f^{s})$: 7107 |  | parametric version of Algorithm 4: 18.98 |
---|---|---|---

  13. $\bullet$ $S_{17}$ :

$f=x^{2}z+yz^{2}+y^{6}+u_{1}y^{4}z+u_{2}z^{3}$

$\operatorname{\mathbb{C}}^{2}\backslash\operatorname{\mathbb{V}}(u_{1})$ | $b_{\text{st}}(s)(s+\frac{23}{24})(s+\frac{27}{24})$
---|---
$\operatorname{\mathbb{V}}(u_{1})\backslash\operatorname{\mathbb{V}}(u_{1},u_{2})$ | $b_{\text{st}}(s)(s+\frac{27}{24})(s+\frac{47}{24})$
$\operatorname{\mathbb{V}}(u_{1},u_{2})$ | $b_{\text{st}}(s)(s+\frac{47}{24})(s+\frac{51}{24})$

$b_{\text{st}}(s)=(s+\frac{21}{24})(s+\frac{25}{24})(s+\frac{28}{24})(s+\frac{29}{24})(s+\frac{31}{24})(s+\frac{32}{24})(s+\frac{33}{24})(s+\frac{35}{24})$

$\times(s+\frac{36}{24})(s+\frac{37}{24})(s+\frac{39}{24})(s+\frac{40}{24})(s+\frac{41}{24})(s+\frac{43}{24})(s+\frac{44}{24}).$

CGS of $\text{Ann}(f^{s})$: 26220 |  | parametric version of Algorithm 4: 3.063 |
---|---|---|---

  14. $\bullet$ $U_{16}$:

$x^{3}+xz^{2}+y^{5}+u_{1}y^{2}z^{2}+u_{2}y^{3}z^{2}$

$\operatorname{\mathbb{C}}^{2}\backslash\operatorname{\mathbb{V}}(u_{1})$ | $b_{\text{st}}(s)(s+\frac{14}{15})(s+\frac{17}{15})$
---|---
$\operatorname{\mathbb{V}}(u_{1})\backslash\operatorname{\mathbb{V}}(u_{1},u_{2})$ | $b_{\text{st}}(s)(s+\frac{17}{15})(s+\frac{29}{15})$
$\operatorname{\mathbb{V}}(u_{1},u_{2})$ | $b_{\text{st}}(s)(s+\frac{29}{15})(s+\frac{32}{15})$

$b_{\text{st}}(s)=(s+\frac{13}{15})(s+\frac{16}{15})(s+\frac{18}{15})(s+\frac{19}{15})(s+\frac{21}{15})(s+\frac{22}{15})(s+\frac{23}{15})(s+\frac{24}{15})(s+\frac{26}{15})(s+\frac{27}{15})$

CGS of $\text{Ann}(f^{s})$: 3141 |  | parametric version of Algorithm 4: 5.156 |
---|---|---|---

  15. $\bullet$ $J_{16}$:

$f=x^{3}+y^{9}+u_{1}x^{2}y^{3}+u_{2}y^{10}$ ($4u_{1}^{3}+27\neq 0$)

stratum | $\tilde{b}_{f,0}(s)$ | note
---|---|---
$\operatorname{\mathbb{C}}^{2}\backslash\operatorname{\mathbb{V}}((4u_{1}^{3}+27)u_{1}u_{2})$ | $b_{\text{st}}(s)$ | $\dim_{\operatorname{\mathbb{C}}}(H_{M_{(-5/9,f)}})=2$
$\operatorname{\mathbb{V}}(u_{1}u_{2})\backslash\operatorname{\mathbb{V}}(4u_{1}^{3}+27)$ | $b_{\text{st}}(s)(s+\frac{14}{9})$ | $\dim_{\operatorname{\mathbb{C}}}(H_{M_{(-5/9,f)}})=1,$
|  | $\dim_{\operatorname{\mathbb{C}}}(H_{M_{(-14/9,f)}})=1$

$b_{\text{st}}(s)=(s+\frac{4}{9})(s+\frac{5}{9})(s+\frac{6}{9})(s+\frac{7}{9})(s+\frac{8}{9})(s+\frac{9}{9})(s+\frac{10}{9})(s+\frac{11}{9})(s+\frac{12}{9})(s+\frac{13}{9}).$

CGS of $\text{Ann}(f^{s})$: 55.83 |  | parametric version of Algorithm 4: 304.6 |
---|---|---|---

  16. $\bullet$ $W_{15}$ :

$f=x^{4}+y^{6}+u_{1}x^{2}y^{3}+u_{2}y^{7}$ ($u_{1}^{2}-4\neq 0$)

$\operatorname{\mathbb{C}}^{2}\backslash\operatorname{\mathbb{V}}((u_{1}^{2}-4)u_{1}u_{2})$ | $b_{\text{st}}(s)$ | $\dim_{\operatorname{\mathbb{C}}}(H_{M_{(-7/12,f)}})=2$
---|---|---
$\operatorname{\mathbb{V}}(u_{1}u_{2})\backslash\operatorname{\mathbb{V}}(u_{1}^{2}-4)$ | $b_{\text{st}}(s)(s+\frac{19}{12})$ | $\dim_{\operatorname{\mathbb{C}}}(H_{M_{(-7/12,f)}})=1,$
|  | $\dim_{\operatorname{\mathbb{C}}}(H_{M_{(-19/12,f)}})=1$

$b_{\text{st}}(s)=(s+\frac{5}{12})(s+\frac{7}{12})(s+\frac{8}{12})(s+\frac{9}{12})(s+\frac{10}{12})(s+\frac{11}{12})(s+\frac{12}{12})(s+\frac{13}{12})$

$\times(s+\frac{14}{12})(s+\frac{15}{12})(s+\frac{16}{12})(s+\frac{17}{12}).$

CGS of $\text{Ann}(f^{s})$: 16.88 |  | parametric version of Algorithm 4: 158 |
---|---|---|---

  17. $\bullet$ $Z_{15}$ :

$f=x^{3}y+y^{7}+u_{1}x^{2}y^{3}+u_{2}y^{8}$ ($4u_{1}^{3}+27\neq 0$)

$\operatorname{\mathbb{C}}^{2}\backslash\operatorname{\mathbb{V}}((4u_{1}^{3}+27)u_{1}u_{2})$ | $b_{\text{st}}(s)$ | $\dim_{\operatorname{\mathbb{C}}}(H_{M_{(-4/7,f)}})=2$
---|---|---
$\operatorname{\mathbb{V}}(u_{1}u_{2})\backslash\operatorname{\mathbb{V}}(4u_{1}^{3}+27)$ | $b_{\text{st}}(s)(s+\frac{11}{7})$ | $\dim_{\operatorname{\mathbb{C}}}(H_{M_{(-4/7,f)}})=1$
|  | $\dim_{\operatorname{\mathbb{C}}}(H_{M_{(-11/7,f)}})=1$

$b_{\text{st}}(s)=(s+\frac{3}{7})(s+\frac{4}{7})(s+\frac{5}{7})(s+\frac{6}{7})(s+\frac{7}{7})(s+\frac{8}{7})(s+\frac{9}{7})(s+\frac{10}{7}).$

CGS of $\text{Ann}(f^{s})$: 36.86 |  | parametric version of Algorithm 4: 767.2 |
---|---|---|---

  18. $\bullet$ $Q_{14}$:

$f=x^{3}+yz^{2}+u_{1}x^{2}y^{2}+xy^{4}+u_{2}y^{7}$ ($u_{1}^{2}-4\neq 0$)

$\operatorname{\mathbb{C}}^{2}\backslash\operatorname{\mathbb{V}}((u_{1}^{2}-4)(4u_{1}^{2}-15)u_{2})$ | $b_{\text{st}}(s)$ | $\dim_{\operatorname{\mathbb{C}}}(H_{M_{(-13/12,f)}})=2$
---|---|---
$\operatorname{\mathbb{V}}(u_{2}(4u_{1}^{2}-15))\backslash\operatorname{\mathbb{V}}(u_{1}^{2}-4)$ | $b_{\text{st}}(s)(s+\frac{25}{12})$ | $\dim_{\operatorname{\mathbb{C}}}(H_{M_{(-13/12,f)}})=1,$
|  | $\dim_{\operatorname{\mathbb{C}}}(H_{M_{(-25/12,f)}})=1$

$b_{\text{st}}(s)=(s+\frac{11}{12})(s+\frac{13}{12})(s+\frac{15}{12})(s+\frac{16}{12})(s+\frac{17}{12})(s+\frac{19}{12})(s+\frac{20}{12})(s+\frac{21}{12})(s+\frac{23}{12}).$

CGS of $\text{Ann}(f^{s})$: 630.8 |  | parametric version of Algorithm 4: 341.6 |
---|---|---|---

  19. $\bullet$ $S_{14}$ :

$f=x^{2}z+yz^{2}+y^{5}+u_{1}y^{3}z+u_{2}z^{3}$ ($u_{1}^{2}-4\neq 0$)

$\operatorname{\mathbb{C}}^{2}\backslash\operatorname{\mathbb{V}}((u_{1}^{2}-4)(3u_{1}^{2}-10)u_{2})$ | $b_{\text{st}}(s)$ | $\dim_{\operatorname{\mathbb{C}}}(H_{M_{(-11/10,f)}})=2$
---|---|---
$\operatorname{\mathbb{V}}(u_{2}(3u_{1}^{2}-10))\backslash\operatorname{\mathbb{V}}(u_{1}^{2}-4)$ | $b_{\text{st}}(s)(s+\frac{21}{10})$ | $\dim_{\operatorname{\mathbb{C}}}(H_{M_{(-11/10,f)}})=1$
|  | $\dim_{\operatorname{\mathbb{C}}}(H_{M_{(-21/10,f)}})=1$

$b_{\text{st}}(s)=(s+\frac{9}{10})(s+\frac{11}{10})(s+\frac{12}{10})(s+\frac{13}{10})(s+\frac{14}{10})(s+\frac{15}{10})(s+\frac{16}{10})(s+\frac{17}{10})(s+\frac{18}{10})(s+\frac{19}{10}).$

CGS of $\text{Ann}(f^{s})$: 295.1 |  | parametric version of Algorithm 4: 7.438 |
---|---|---|---

  20. $\bullet$ $U_{14}$ :

$f=x^{3}+xz^{2}+u_{1}xy^{3}+y^{3}z+u_{2}yz^{3}$ ($u_{1}^{2}+1\neq 0$)

$\operatorname{\mathbb{C}}^{2}\backslash\operatorname{\mathbb{V}}((u_{1}^{2}+1)(2u_{1}^{2}+3)u_{2})$ | $b_{\text{st}}(s)$ | $\dim_{\operatorname{\mathbb{C}}}(H_{M_{(-10/9,f)}})=2$
---|---|---
$\operatorname{\mathbb{V}}(u_{2}(2u_{1}^{2}+3))\backslash\operatorname{\mathbb{V}}(u_{1}^{2}+1)$ | $b_{\text{st}}(s)(s+\frac{19}{9})$ | $\dim_{\operatorname{\mathbb{C}}}(H_{M_{(-10/9,f)}})=1,$
|  | $\dim_{\operatorname{\mathbb{C}}}(H_{M_{(-19/9,f)}})=1$

$b_{\text{st}}(s)=(s+\frac{8}{9})(s+\frac{10}{9})(s+\frac{11}{9})(s+\frac{12}{9})(s+\frac{13}{9})(s+\frac{14}{9})(s+\frac{15}{9})(s+\frac{16}{9})(s+\frac{17}{9}).$

CGS of $\text{Ann}(f^{s})$: 21230 |  | parametric version of Algorithm 4: 304.3 |
---|---|---|---

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  * [8] M. Kato,  The $b$-function of $\mu$-constant deformation of $x^{7}+y^{5}$,  Bull. College of Science, Univ. of the Ryukyus 32 (1981), 5–10.
  * [9] M. Kato,  The $b$-function of $\mu$-constant deformation of $x^{9}+y^{4}$,  Bull. College of Science, Univ. of the Ryukyus 33 (1982), 5–8.
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# Global Lipschitz stability for an inverse coefficient problem for a mean
field game system

1 Oleg Imanuvilov and 2 Masahiro Yamamoto

###### Abstract

For an inverse coefficient problem of determining a state-varying factor in
the corresponding Hamiltonian for a mean field game system, we prove the
global Lipschitz stability by spatial data of one component and interior data
in an arbitrarily chosen subdomain over a time interval. The proof is based on
Carleman estimates with different norms.

1 Department of Mathematics, Colorado State University, 101 Weber Building,
Fort Collins CO 80523-1874, USA e-mail<EMAIL_ADDRESS>
2 Graduate School of Mathematical Sciences, The University of Tokyo, Komaba,
Meguro, Tokyo 153-8914, Japan e-mail<EMAIL_ADDRESS>
##  1 Introduction

Let $\Omega\subset\mathbb{R}^{d}$ be a bounded domain with smooth boundary
$\partial\Omega$, and let $T>0$, and $Q:=\Omega\times(0,T)$. In article
concerned a system of the mean field game:

$\left\\{\begin{array}[]{rl}&\partial_{t}u(x,t)+\Delta
u(x,t)-\frac{1}{2}p(x)|\nabla u(x,t)|^{2}-c_{0}(x)v(x,t)=0\quad\mbox{in
$Q$},\\\ &\partial_{t}v(x,t)-\Delta
v(x,t)-\mbox{div}\thinspace(p(x)v(x,t)\nabla u(x,t))=0\quad\mbox{in $Q$},\\\
&u|_{\partial\Omega\times(0,T)}=v|_{\partial\Omega\times(0,T)}=0.\end{array}\right.$
$None$

In (1.1), $x$ and $t$ are the state and the time variables, and $u$ and $v$
denote the value of the game and the population density of players
respectively (e.g., [1], [12]). We note that $p(x)$ specifies the Hamiltonian
for (1.1).

We consider
Inverse coefficient problem. Let $\omega\subset\Omega$ be an arbitrarily
chosen subdomain and $t_{0}\in(0,T)$, $\delta_{0}>0$ be small. One need to
determine $p(x)$, $x\in\Omega$ by
$u|_{\omega\times(t_{0}-\delta_{0}\,t_{0}+\delta_{0})}$,
$v|_{\omega\times(t_{0}-\delta_{0}\,t_{0}+\delta_{0})}$ and $u(\cdot,t_{0})$
in $\Omega$.

Henceforth we set $\partial_{i}:=\frac{\partial}{\partial x_{i}}$ for $1\leq
i\leq d$, and define $H^{2,1}(Q):=\\{u\in L^{2}(Q);\,\nabla
u,\partial_{i}\partial_{j}u,\partial_{t}u\in L^{2}(Q),\,\,1\leq i,j\leq d\\}$,
and $W^{1,\infty}(\Omega)$ denotes the Sobolev space of functions whose first
partial derivatives are in $L^{\infty}(\Omega).$

We state our main result.
Theorem 1. For $\ell=1,2$, let $(u_{\ell},v_{\ell})\in(H^{2,1}(Q))^{2}$
satisfy (1.1) with $p:=p_{\ell}$. We assume that

$\left\\{\begin{array}[]{rl}&\|p_{\ell}\|_{W^{1,\infty}(\Omega)}\leq
M,\quad\|c_{0}\|_{L^{\infty}(\Omega)}\leq M,\\\
&\|u_{\ell}\|_{W^{1,\infty}(Q)}+\|\partial_{t}u_{\ell}\|_{L^{\infty}(0,T;W^{1,\infty}(\Omega))}\leq
M,\,\|v_{\ell}\|_{W^{1,\infty}(Q)}+\|\partial_{t}v_{\ell}\|_{L^{\infty}(0,T;W^{1,\infty}(\Omega))}\leq
M,\\\
&(\partial_{t}^{k}u_{\ell},\partial_{t}^{k}v_{\ell})\in(H^{2,1}(Q))^{2},\quad
k=0,1,\,\ell=1,2\end{array}\right.$ $None$

and there exists a constant $\delta>0$ such that

$|\nabla u_{1}(x,t_{0})|\geq\delta,\,\,x\in\Omega\quad\mbox{or}\quad|\nabla
u_{2}(x,t_{0})|\geq\delta,\,\,x\in\Omega.$

Then there exists a constant $C=C(M,\delta)>0$ such that

$\|p_{1}-p_{2}\|_{L^{2}(\Omega)}\leq
C(\|u_{1}(\cdot,t_{0})-u_{2}(\cdot,t_{0})\|_{H^{2}(\Omega)}+\|u_{1}-u_{2}\|_{H^{1}(0,T;L^{2}(\omega))}+\|v_{1}-v_{2}\|_{H^{1}(0,T;L^{2}(\omega))}).$

We emphasize the following features of Theorem 1:

  1. (1)

The observation subdomain $\omega\subset\Omega$ can be arbitrarily small.

  2. (2)

Lipschitz stability over $\Omega$.

  3. (3)

We need the positiveness of $|\nabla u_{1}|$ or $|\nabla u_{2}|$ only at one
moment $t=t_{0}$.

  4. (4)

We do not need spatial data neither $v_{1}$ nor $v_{2}$ in
$\Omega\times\\{t_{0}\\}$.

The features (1) - (3) are inevitable consequences of our methodology (e.g.,
[6]). The last (4) is a new aspect by that the unknown is only one spatial
function, so that we do not need $v(\cdot,t_{0})$ but one spatial data are
sufficient. We remark that the stability and the uniqueness are open in the
cases of $t_{0}=0$ and $t_{0}=T$.

As for inverse problems for mean field games, see [3], [4], [7] \- [11], [13]
\- [16]. In particular, [8] proves Hölder stability with extra data compare to
the above inverse problem for the case where the equation contain special non-
local term.

The proof of Theorem 1 is based on Carleman estimates: Lemma 2 in Section 2
and the modified argument in [6].

##  2 Main Carleman estimate

Setting $y:=u_{1}-u_{2}$, $z:=v_{1}-v_{2}$ and $f:=\frac{1}{2}(p_{1}-p_{2})$,
$g:=|\nabla u_{1}|^{2}$, $h:=2v_{1}\nabla u_{1}$,
$r_{1}:=\frac{1}{2}p_{2}(\nabla u_{1}+\nabla u_{2})$, $r_{2}:=p_{2}\nabla
u_{1}$ and $r_{3}:=p_{2}v_{2}.$ From (1.2) we obtain the linearized system:

$\left\\{\begin{array}[]{rl}&\partial_{t}y(x,t)+\Delta
y(x,t)=c_{0}z+r_{1}\cdot\nabla y+g(x,t)f(x)\quad\mbox{in $Q$},\\\
&\partial_{t}z(x,t)-\Delta z(x,t)=\mbox{div}\thinspace(r_{2}z+r_{3}\nabla
y)+\mbox{div}\thinspace(h(x,t)f(x))\quad\mbox{in $Q$}\end{array}\right.$
$None$

with $y=z=0$ on $\partial\Omega\times(0,T)$. By (1.2) we see that

$\partial_{t}^{j}r_{k},\partial_{t}^{j}g,\partial_{t}^{j}h\in
L^{\infty}(Q)\quad\mbox{with $1\leq k\leq 3$ and $j=0,1$}.$ $None$

For the Carleman estimates, we introduce $\eta\in C^{2}(\overline{\Omega})$
such that $\eta>0$ in $\Omega$, $\eta|_{\partial\Omega}=0$ and
$|\nabla\eta|>0$ on $\overline{\Omega\setminus\omega_{0}}$, where $\omega_{0}$
is some subdomain such that $\overline{\omega_{0}}\subset\omega$. (For
existence of such a function see e.g. [2].) Without loss of generality, we can
assume that $t_{0}=\frac{1}{2}T$ and
$(t_{0}-\delta_{0}\,t_{0}+\delta_{0})=(0,T)$.

Fixing a constant $\lambda>0$ sufficiently large, we set $\mu(t):=t(T-t)$,
$Q_{\omega}:=\omega\times(0,T)$, and

$\varphi(x,t):=\frac{e^{\lambda\eta(x)}}{\mu(t)},\quad\alpha(x,t):=\frac{e^{\lambda\eta(x)}-e^{2\lambda\|\eta\|_{C(\overline{\Omega})}}}{\mu(t)},\quad(x,t)\in
Q.$

Then we state Carleman estimates for single parabolic equations, which can be
both backward and forward.
Lemma 1. Let $\widetilde{y},\widetilde{z}\in H^{2,1}(Q)$ and
$\widetilde{y}|_{\partial\Omega\times(0,T)}=\widetilde{z}|_{\partial\Omega\times(0,T)}=0$.
There exist constants $s_{0}>0$ and $C>0$ such that for all $s>s_{0}$, we have

$\int_{Q}\left(\frac{1}{s\varphi}\left(\sum_{i,j=1}^{d}|\partial_{i}\partial_{j}\widetilde{y}|^{2}+|\partial_{t}\widetilde{y}|^{2}\right)+s\varphi|\nabla\widetilde{y}|^{2}+s^{3}\varphi^{3}|\widetilde{y}|^{2}\right)e^{2s\alpha}dxdt$
$\leq
C\int_{Q}|\partial_{t}\widetilde{y}+\Delta\widetilde{y}|^{2}e^{2s\alpha}dxdt+C\int_{Q_{\omega}}s^{3}\varphi^{3}|\widetilde{y}|^{2}e^{2s\alpha}dxdt$
$None$

and

$\int_{Q}\left(\frac{1}{s\varphi}|\nabla\widetilde{z}|^{2}+s\varphi|\widetilde{z}|^{2}\right)e^{2s\alpha}dxdt\leq
C\int_{Q}|G|^{2}e^{2s\alpha}dxdt+C\int_{Q_{\omega}}s\varphi|\widetilde{z}|^{2}e^{2s\alpha}dxdt$
$None$

where $\partial_{t}\widetilde{z}-\Delta\widetilde{z}=\mbox{div}\thinspace G$
in $Q$.

The Carleman estimate (2.3) can be proved by applying
$\alpha(x,t)=\alpha(x,T-t)$ and $\varphi(x,t)=\varphi(x,T-t)$ for $(x,t)\in Q$
to the Carleman estimate in [2], while the proof of (2.4) is found in
Imanuvilov and Yamamoto [5].

Henceforth $C>0$ denotes generic constants independent of $s>0$. We define
$D(y,z):=\int_{Q_{\omega}}(s^{3}\varphi^{3}|y|^{2}+s\varphi|z|^{2})e^{2s\alpha}dxdt$.
Setting $y_{1}:=\partial_{t}y$ and $z_{1}:=\partial_{t}z$, and differentiating
equations (2.1) respect to $t$ we have

$\left\\{\begin{array}[]{rl}&\partial_{t}y_{1}+\Delta
y_{1}=c_{0}z_{1}+r_{1}\cdot\nabla y_{1}+(\partial_{t}r_{1})\cdot\nabla
y+(\partial_{t}g)f\quad\mbox{in $Q$},\\\ &\partial_{t}z_{1}-\Delta
z_{1}=\mbox{div}\thinspace(r_{2}z_{1}+r_{3}\nabla
y_{1})+\mbox{div}\thinspace((\partial_{t}r_{2})z+(\partial_{t}r_{3})\nabla
y)+\mbox{div}\thinspace((\partial_{t}h)f)\quad\mbox{in
$Q$}.\end{array}\right.$ $None$

Applying (2.3) and (2.4) to the first and the second equations in (2.1)
respectively, in terms of (2.2) we have

$\int_{Q}\left(\frac{1}{s\varphi}|\partial_{t}y|^{2}+s\varphi|\nabla
y|^{2}+s^{3}\varphi^{3}|y|^{2}\right)e^{2s\alpha}dxdt\leq
C\int_{Q}|z|^{2}e^{2s\alpha}dxdt+C\int_{Q}|f|^{2}e^{2s\alpha}dxdt+CD(y,z)$
$None$

and

$\int_{Q}s\varphi|z|^{2}e^{2s\alpha}dxdt\leq C\int_{Q}(|y|^{2}+|\nabla
y|^{2})e^{2s\alpha}dxdt+C\int_{Q}|f|^{2}e^{2s\alpha}dxdt+CD(y,z).$ $None$

Here in terms of (2.2), we estimate $\int_{Q}|r_{1}\cdot\nabla
y|^{2}e^{2s\alpha}dxdt\leq C\int_{Q}|\nabla y|^{2}e^{2s\alpha}dxdt$, and this
term can be absorbed into $\int_{Q}s\varphi|\nabla y|^{2}e^{2s\alpha}dxdt$ on
the left-hand side of (2.3) for large $s>0$. Throughout the proof, we repeat
similar estimation with absorption thanks to the large parameter $s>0$.

Adding (2.6) and (2.7), and choosing $s>0$ large, we can absorb the resulting
term on the right-hand side into the left-hand side, so that

$\int_{Q}\left(\frac{1}{s\varphi}|\partial_{t}y|^{2}+s\varphi|\nabla
y|^{2}+s^{3}\varphi^{3}|y|^{2}+s\varphi|z|^{2}\right)e^{2s\alpha}dxdt\leq
C\int_{Q}|f|^{2}e^{2s\alpha}dxdt+CD(y,z).$ $None$

Next, the application of Lemma 1 to (2.5) yields

$\int_{Q}\left(\frac{1}{s\varphi}|\partial_{t}y_{1}|^{2}+s\varphi|\nabla
y_{1}|^{2}+s^{3}\varphi^{3}|y_{1}|^{2}\right)e^{2s\alpha}dxdt$ $\leq
C\int_{Q}(|\nabla
y|^{2}+|z_{1}|^{2})e^{2s\alpha}dxdt+C\int_{Q}|f|^{2}e^{2s\alpha}dxdt+CD(y_{1},z_{1})$
$None$

and

$\int_{Q}s\varphi|z_{1}|^{2}e^{2s\alpha}dxdt\leq C\int_{Q}(|z|^{2}+|\nabla
y|^{2}+|\nabla
y_{1}|^{2})e^{2s\alpha}dxdt+C\int_{Q}|f|^{2}e^{2s\alpha}dxdt+CD(y_{1},z_{1}).$
$None$

Adding (2.9) and (2.10), we can absorb the terms $|z_{1}|^{2}$, $|\nabla
y_{1}|^{2}$ on the right-hand side into the left-hand side, we can obtain

$\displaystyle\int_{Q}\left(\frac{1}{s\varphi}|\partial_{t}y_{1}|^{2}+s\varphi|\nabla
y_{1}|^{2}+s^{3}\varphi^{3}|y_{1}|^{2}+s\varphi|z_{1}|^{2}\right)e^{2s\alpha}dxdt$
$\displaystyle\leq$ $\displaystyle C\int_{Q}(|\nabla
y|^{2}+|z|^{2})e^{2s\alpha}dxdt+C\int_{Q}|f|^{2}e^{2s\alpha}dxdt+CD(y_{1},z_{1}).$

Substituting (2.8) into the first term on the right-hand side, we reach
Lemma 2 (key Carleman estimate. There exist constants $s_{0}>0$ and $C>$ such
that

$\int_{Q}\biggl{(}\frac{1}{s\varphi}|\partial_{t}^{2}y|^{2}+s^{3}\varphi^{3}|\partial_{t}y|^{2})+s\varphi(|z|^{2}+|\partial_{t}z|^{2})\biggr{)}e^{2s\alpha}dxdt\leq
C\int_{Q}|f|^{2}e^{2s\alpha}dxdt+C(D(y,z)+D(y_{1},z_{1}))$

for all $s>s_{0}$.

##  3 Completion of the proof of Theorem 1.

By $e^{2s\alpha(x,0)}=0$ for $x\in\Omega$, and $|\partial_{t}\varphi|\leq
C\varphi^{2},|\partial_{t}\alpha|\leq C\varphi^{2}$ in $Q$, we have

$\displaystyle\int_{\Omega}\varphi(x,t_{0})^{-1}|\partial_{t}y(x,t_{0})|^{2}e^{2s\alpha(x,t_{0})}dx=\int^{t_{0}}_{0}\frac{d}{dt}\left(\int_{\Omega}\varphi^{-1}|\partial_{t}y|^{2}e^{2s\alpha}dx\right)dt$
$\displaystyle=$
$\displaystyle\int^{t_{0}}_{0}\int_{\Omega}(-(\partial_{t}\varphi)\varphi^{-2}|\partial_{t}y|^{2}+2s\varphi^{-1}|\partial_{t}y|^{2}(\partial_{t}\alpha)+2\varphi^{-1}(\partial_{t}y)(\partial_{t}^{2}y))e^{2s\alpha}dxdt$
$\displaystyle\leq$ $\displaystyle
C\int_{Q}(|\partial_{t}y|^{2}+s\varphi|\partial_{t}y|^{2}+|\partial_{t}y||\partial_{t}^{2}y|)e^{2s\alpha}dxdt\leq
C\int_{Q}(s\varphi|\partial_{t}y|^{2}+|\partial_{t}y||\partial_{t}^{2}y|)e^{2s\alpha}dxdt$
$\displaystyle\leq$ $\displaystyle
C\int_{Q}(s\varphi|\partial_{t}y|^{2}+\frac{1}{s\varphi}|\partial_{t}^{2}y|^{2})e^{2s\alpha}dxdt.$

Here we used
$|\partial_{t}y||\partial_{t}^{2}y|=\left(\frac{1}{\sqrt{s\varphi}}|\partial_{t}^{2}y|\right)(\sqrt{s\varphi}|\partial_{t}y|)\leq\frac{1}{2}\left(\frac{1}{s\varphi}|\partial_{t}^{2}y|^{2}+s\varphi|\partial_{t}y|^{2}\right)$.
Therefore, by $\min_{x\in\overline{\Omega}}\varphi(x,t_{0})^{-1}>0$, Lemma 2
yields

$\int_{\Omega}|\partial_{t}y(x,t_{0})|^{2}e^{2s\alpha(x,t_{0})}dx\leq
C\int_{\Omega}\varphi(x,t_{0})^{-1}|\partial_{t}y(x,t_{0})|^{2}e^{2s\alpha(x,t_{0})}dx\leq
C\int_{Q}|f|^{2}e^{2s\alpha}dxdt+C_{s}\mathcal{D}.$

Here and henceforth we set
$\mathcal{D}:=\|y\|^{2}_{H^{1}(0,T;L^{2}(\omega))}+\|z\|^{2}_{H^{1}(0,T;L^{2}(\omega))}$.

We can assume that $|\nabla u_{1}(x,t_{0})|>0$ for $x\in\overline{\Omega}$.
Then, since $|g(x,t_{0})|=|\nabla u_{1}(x,t_{0})|^{2}>0$ for
$x\in\overline{\Omega}$ and $g(x,t_{0})f(x)=\partial_{t}y(x,t_{0})+(\Delta
y-r_{1}\nabla y)(x,t_{0})-c_{0}(x)z(x,t_{0})$ for $x\in\Omega$, we obtain

$\int_{\Omega}|f(x)|^{2}e^{2s\alpha(x,t_{0})}dx\leq
C\int_{Q}|f|^{2}e^{2s\alpha}dxdt+C_{s}\mathcal{D}+C_{s}\|y(\cdot,t_{0})\|^{2}_{H^{2}(\Omega)}+C\int_{\Omega}|z(x,t_{0})|^{2}e^{2s\alpha(x,t_{0})}dx.$
$None$

Next

$\displaystyle\int_{\Omega}|z(x,t_{0})|^{2}e^{2s\alpha(x,t_{0})}dx\leq
C\int_{\Omega}\varphi(x,t_{0})^{-1}|z(x,t_{0})|^{2}e^{2s\alpha(x,t_{0})}dx$
$\displaystyle=$ $\displaystyle
C\int^{t_{0}}_{0}\frac{d}{dt}\left(\int_{\Omega}\varphi^{-1}|z|^{2}e^{2s\alpha}dx\right)dt$
$\displaystyle=$ $\displaystyle
C\int^{t_{0}}_{0}\int_{\Omega}(-(\partial_{t}\varphi)\varphi^{-2}|z|^{2}+\varphi^{-1}|z|^{2}2s(\partial_{t}\alpha)+\varphi^{-1}2z(\partial_{t}z))e^{2s\alpha}dxdt$
$\displaystyle\leq$ $\displaystyle
C\int_{Q}(s\varphi|z|^{2}+|z||\partial_{t}z|)e^{2s\alpha}dxdt\leq
C\int_{Q}(s\varphi|z|^{2}+|z|^{2}+|\partial_{t}z|^{2})e^{2s\alpha}dxdt.$

Therefore Lemma 2 yields
$\int_{\Omega}|z(x,t_{0})|^{2}e^{2s\alpha(x,t_{0})}dx\leq
C\int_{Q}|f|^{2}e^{2s\alpha}dxdt+C_{s}\mathcal{D}$, with which (3.1) implies

$\int_{\Omega}|f(x)|^{2}e^{2s\alpha(x,t_{0})}dx\leq
C\int_{Q}|f|^{2}e^{2s\alpha}dxdt+C_{s}(\mathcal{D}+\|y(\cdot,t_{0})\|^{2}_{H^{2}(\Omega)})$
$None$

for all large $s>0$.

On the other hand,

$\int_{Q}|f(x)|^{2}e^{2s\alpha(x,t)}dxdt=\int_{\Omega}|f(x)|^{2}e^{2s\alpha(x,t_{0})}\left(\int^{T}_{0}e^{2s(\alpha(x,t)-\alpha(x,t_{0}))}dt\right)dx.$

Since $\mu(t_{0})>\mu(t)$ for $t\neq t_{0}$, we verify
$\alpha(x,t_{0})-\alpha(x,t)\geq
C_{0}\left(\frac{1}{\mu(t)}-\frac{1}{\mu(t_{0})}\right)$ for $(x,t)\in Q$,
where
$C_{0}:=e^{2\lambda\|\eta\|_{C(\overline{\Omega})}}-e^{\lambda\|\eta\|_{C(\overline{\Omega})}}$.

Hence,

$\int^{T}_{0}e^{2s(\alpha(x,t)-\alpha(x,t_{0}))}dt\leq\int^{T}_{0}\exp\left(-2sC_{0}\left(\frac{1}{\mu(t)}-\frac{1}{\mu(t_{0})}\right)\right)dt$

for $x\in\Omega$. Since
$\lim_{s\to\infty}\exp\left(-2sC_{0}\left(\frac{1}{\mu(t)}-\frac{1}{\mu(t_{0})}\right)\right)=0$
if $t\neq t_{0}$ and
$\exp\left(-2sC_{0}\left(\frac{1}{\mu(t)}-\frac{1}{\mu(t_{0})}\right)\right)\leq
1$ for $s>0$ and $0\leq t\leq T$, the Lebesgue convergence theorem yields
$\sup_{x\in\Omega}\int^{T}_{0}e^{2s(\alpha(x,t)-\alpha(x,t_{0}))}dt=o(1)$ as
$s\to\infty$, and so
$\int_{Q}|f|^{2}e^{2s\alpha}dxdt=o(1)\int_{\Omega}|f|^{2}e^{2s\alpha(x,t_{0})}dx$.
Substituting this into (3.2) and choosing $s>0$ large, we reach
$\int_{\Omega}|f|^{2}e^{2s\alpha(x,t_{0})}dx\leq
C_{s}(\mathcal{D}+\|y(\cdot,t_{0})\|^{2}_{H^{2}(\Omega)})$. $\blacksquare$

Acknowledgments. The work was supported by Grant-in-Aid for Scientific
Research (A) 20H00117 and Grant-in-Aid for Challenging Research (Pioneering)
21K18142 of Japan Society for the Promotion of Science.

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