数理解析研究所講究録 1233 巻 2001 年 51-66

51
A study of semiquasihomogeneous singularities
by using holonomic system(holonomic 系を用いた半擬斉次孤立特異点の考察)
Yayoi Nakamura, Ochanomizu univ. (中村 弥生, お茶の水女子大学)\dag er
Shinichi Tajima, Niigata univ. (田島 慎一, 新潟大学 )*Introduction

1

In this note, we study the quasihomogeneous singularity and exceptional singularities of modality 1byusing partial differential operators. We examine, in particular, algebraic local cohomology classes attachedto these singularities.
. Let be ain the dimensional affine space
Let $X$ be an open neighborhood of the origin
in the sheaf
ideal
the
by
I
.
Denoteorigintheatsingularity
holomorphic function with an isolated$(j=1, \ldots, n)$ of theof holomorphic functions on $X$ generated by the partial derivatives
$O$

$f$

$\mathbb{C}^{n}$

$n$

$O$

$f_{j}= \frac{\partial f}{\partial z_{j}}$$O_{X}$

function . We denote by. Since the pairing$f$$\Sigma$the set of cohomology classes in$\mathcal{H}_{[0]}^{n}(O_{X})$annihilated by every element in$I$$\Omega_{X,O}/I\Omega_{X,O}\mathrm{x}$(1.1)$\Sigmaarrow \mathbb{C}$defined by the Grothendieck local residue is non-degenerate, )becomes the dual space ofis the sheaf of holomorphic differential -forms on $X$ .
as avector space where
Let abe an algebraic local cohomology class which generates over Ox,o- Since the algebraic local
modules, we can consider annihilators of in thehas a structure ofcohomology group$X$of differential operators on . In this paper, we consider the ideal, denoted by $Ann\leq 1$ , insheaf
generated by annihilators of aof at most first order.
We give the precise definition of $Ann\leq 1$ and we give an description of the solution space of theholonomic system $D_{X}/Ann\leq 1$ in \S 2.
In \S 3, we examine $Ann\leq 1$ in the case of quasihomogeneous singularities. We verify that the coho
mology class attached to aquasihomogeneous singularity can be characterized as the solution of thesystem of differential equations of at most first order (Theorem 3.1).
For non-quasihomogeneous isolated singularities, we show that the dimension of the solution spaceof the holonomic system $D_{X}/Ann\leq 1$ is greater than or equal to 2. Especially, in the case of exceptional
singularities of modality 1, we verify that the dimension of the solution space of $D_{X}/Ann\leq 1$ is just equal
to 2and the basis is given by and the delta function (Theorem 4.1 in \S 4). In \S 4.4, we give results ofcomputations for normal forms of exceptional singularities of modality 1.
$O_{X,O}/I\underline{\simeq}$$n$$\Omega_{X}$$\mathrm{f}\mathrm{t}\mathrm{x},0/\mathrm{I}\mathrm{f}\mathrm{t}\mathrm{x},\mathrm{o}$$\Sigma$$D_{X}$$\mathcal{H}_{[O]}^{n}(O_{X})$$\sigma$$D_{X}$$D_{X}$$\sigma$$\sigma$

2The first order differential operators acting on$\Sigma$

For aholomorphic function $f=f(z_{1}, \ldots, z_{n})$ with an isolated singularity at the origin:
generated by the partial derivatives
ideal in$\mathrm{O}\mathrm{x},\mathrm{o}$$f_{j}= \frac{\partial f}{\partial z_{\mathrm{j}}}(j=1, \ldots,n)$$I=\langle f_{1}, \ldots, f_{n}\rangle_{\mathit{0}}$’nakamura@math.ocha.ac.jp\uparrow tajima@geb.ge. niigata-u.ac.jp.

$O$, let I be the52
Denote by$\Sigma$the set of local cohomology classes annihilated by every element in :$I$$\Sigma=\{\eta\in \mathcal{H}_{[O]}^{n}(O_{X})|g\eta=0, g\in I\}$

Let$\sigma$be agenerator of$\Sigma$over:$\mathrm{O}\mathrm{x},0$$\Sigma=O_{X,O}\sigma$.

.

Let $P$ be apartial differential operator of first order which annihilates the algebraic local cohomology
class . Such an operator $P$ has the following property.
$\sigma$Lemma 2.1 Let be a generator of over OxtO- Let P be a linear partial differential operator of firstorder such that $P\sigma=0$ . Then, we have.
$\Sigma$$\sigma$$P(\Sigma)\subseteq\Sigma$, we can write any
Proof. Since generates over
. Let be the first order part$\Sigma$$\sigma$$h\in Ox,\mathit{0}$$\mathrm{O}\mathrm{x},0$$v_{P}$$\eta\in\Sigma$

aswith some holomorphic function$\eta=h\sigma$.

of the annihilator$\sum_{j=1}^{n}a_{j}(z)\frac{\partial}{\partial z_{\mathrm{j}}}$$P= \sum_{j=1}^{n}a_{j}(z)\frac{\partial}{\partial z_{\mathrm{j}}}+\mathrm{a}\mathrm{j}(\mathrm{z})$We have$P(\eta)$$=$$P(h\sigma)$$=$$(Ph-hP)\sigma+hP\sigma$$=$$v_{P}(h)\sigma\in \mathrm{C}$.

Thus we have
Let be the set of linear partial differential operators of at most first order which annihilate :$P(\Sigma)\subseteq \mathrm{X}.\mathrm{O}$$\mathcal{L}$$\sigma$$\mathcal{L}=\{P=\sum_{\mathrm{j}=1}^{n}a_{j}(z)\frac{\partial}{\partial z_{\mathrm{j}}}+a_{0}(z)|P\sigma=0\}$.

It is obvious from the proof of Lemma 2.1, the condition whether agiven first order differential operator
acts on or not depends only on the first order partof $R$ . We denote by the set of differential.
operators of the form
acting on E. Then,is
if and only if$R$$\Sigma$$\mathcal{V}$$v_{R}$$\sum_{\mathrm{j}=1}^{n}a_{j}(z)\frac{\partial}{\partial z_{j}}$$v= \sum_{\mathrm{j}=1}^{n}a_{j}(z)\frac{\partial}{\partial z_{j}}$$v$satisfies the condition $vg\in I$ for any $g\in I$ , i.e.,$\mathcal{V}=\{v=\sum_{j=1}^{n}a_{j}(z)\frac{\partial}{\partial z_{\mathrm{j}}}|vg\in I^{\forall},g\in I\}$

Lemma 2.2 The mapping, from$\mathcal{L}$to$\mathcal{V}$surjective mapping.

.

$Pmf$ For any.

, uthich associates the first order part$v_{p}\in \mathcal{V}$to$P\in \mathcal{L}$

is

$a$

. Thun we have.
an annihilator
Let us consider the condition that the classbecomes asolution of homogeneous differential
equation $P\eta=0$ for an annihilator of. There eists aholomorphic function
which$v\in \mathcal{V}$, there exists aholomorphic function $h\in O_{X,O}$ such that

$P=v-h\in \mathcal{L}$$v\sigma=\mathrm{h}\mathrm{a}$$\square$$\eta\in\Sigma$$P$satisfies$\eta=h\sigma$$\sigma\in\Sigma$$h\in \mathrm{O}x,0$. Then we have$v_{P}h_{\tau} \equiv\sum_{j=1}^{n}a:(.z)\frac{\partial h}{\partial z_{j}}\in I$
$i$

where
in the form
An element$v_{P}\in \mathcal{V}$part of the differential operator $P$ . It is obvious that to represent, it suffices to take the modulo class mod I of the holomorphic function
induces alinear operator acting on $O_{X,O}/I$ which is also denoted by :is the$\mathrm{f}\mathrm{i}\mathrm{r}\mathrm{s}\mathrm{t}’ \mathrm{o}\mathrm{r}\mathrm{d}\mathrm{e}\mathrm{r}$$h$$\eta=\mathrm{h}\mathrm{a}$$v\in \mathcal{V}$

$v$

$v$

: $O_{X,O}/Iarrow O_{X,O}/I$ .

We can put$\mathcal{H}=\{h\in \mathrm{O}\mathrm{x},0/\mathrm{I} |vh=0, \forall v\in \mathcal{V}\}$

Put $Ann\leq 1=D_{X}\mathcal{L}$ . $Ann\leq 1$ defines aleft ideal in$D_{X}$$\eta\in\Sigma$$h\in \mathrm{O}\mathrm{x},0$.

. We have the next theorem.

53
Theorem 2.1$Hom_{D_{X}}$ $(Dx/Ann\leq 1,\mathcal{H}_{[O]}^{n}(O_{X}))=\mathrm{S}\mathrm{p}\mathrm{a}\mathrm{n}\{\mathrm{k}| h\in \mathcal{H}\}$

Proof. Since$I\subset Ann\leq 1$.

as an ideal of multiplicative operators, we have
$Hm\iota_{D_{X}}(D_{X}/Ann\leq 1,\mathcal{H}_{[O]}^{n}(O_{X}))\subset\Sigma$.

Thus we can write any solution of the holonomic system $D_{X}/Ann\leq 1$ as, we have$h\sigma$for some$h\in Ox,\mathit{0}$. For$P\in \mathcal{L}$$P(h\sigma)=v_{P}(h)\sigma=0$Thus we have $v_{P}h=0$ .

.

$\square$

3The case of quasihomogeneous singularities
Let be agenerator of over $O_{X,O}$ . Let Ann be aleft ideal in
algebraic local cohomology class .
$\Sigma$$\sigma$$D_{X}$

consisting of annihilators of the$\sigma$

Theorem 3.1 The following three conditions are equivalent:(i)$\mathit{0}_{X,O}\langle f, f_{1}, \cdots, f_{n}\rangle=O_{X,O}\langle f_{1}, \cdots, f_{n}\rangle$, where$f_{j}:= \frac{\partial f}{\partial z_{j}},$$\int\dot{=}1$,$\ldots$, .
$n$(ii) $Ann\leq 1=Ann$ .

(iii)

.

$Hom_{D_{X}}(D_{X}/Ann\leq 1,\mathcal{H}_{[O]}^{n}(Ox))=\mathrm{S}\mathrm{p}\mathrm{a}\mathrm{n}\{\sigma\}$

Proof.

: Suppose thatcan be expressed in terms of the derivatives $/i$ ,for $f\in Ox,0$ . Then the function$O_{X,\mathit{0}}\langle f, f1, \cdots, f_{n}\rangle=O_{X,O}\langle f_{1}, \cdots, f_{n}\rangle$$(\mathrm{i})\Rightarrow(\mathrm{i}\mathrm{i})$$\ldots$,$f_{n}$. We have

$f$

(3.1)

f
$==a_{1}f \cdots+a_{n}f_{n}a_{1}\frac{\partial f1+}{\partial z_{1}}+\cdots+a_{n}\frac{\partial f}{\partial z_{n}}$,$=(a_{1} \frac{\partial}{\partial z_{1}}+\cdots+a_{n}\frac{\partial}{\partial z_{n}})f$with$a_{1}$,$\ldots$, $a_{n}\in O_{X,O}$ . Assume that ( ,$a_{1}$, an)\neq $(0, \ldots,0)$ . Put$\ldots$(3.1), we have $f_{j}=(a_{1j}f1+\cdots+a_{nj}f_{n})+vfj$ where

we have$v\in \mathcal{V}$$a_{0}\in O_{X,\mathit{0}}$$a_{kj}= \frac{\partial a_{k}}{\partial z_{\mathrm{j}}}$$v=a_{1} \frac{\partial}{\partial z_{1}}+\cdots+a_{n}\frac{\partial}{\partial z_{n}}$.

$i^{\mathrm{R}\mathrm{o}\mathrm{m}}$. As $vfj=fj-(a_{1j}f_{1}+\cdots+a_{nj}f_{n})\in I$ ,. iRom Lemma 2.2, we have an annihilator $P=v+a_{0}$ of the cohomology class afor some

. We have($f_{1}$,$\ldots$,$f_{n}$,$P\rangle$$\subseteq Ann\leq 1\subseteq Ann$.

,is not zero at the origin. This assures that the holonomic
It is known in [2] that the Jacobian of ,, , $P$ ) becomes simple. Since the holonomic system $Dx/Ann$ is simple, we have,system$a_{1}$$D_{X}/\langle f_{1}$$\ldots$$a_{n}$$\ldots$$f_{n}$.

$\langle f_{1}, \ldots, f_{n}, P\rangle=Ann\leq 1=Ann$$(\mathrm{i}\mathrm{i}\mathrm{i})\Rightarrow(\mathrm{i})$: Assume that$f\not\in Ox,0\langle f1, \ldots, f_{n}\rangle$

the multiplicative operator defined by. Let us denote by $F\in Dx$. Obviously, we have
. If the differential operator$f\sigma\neq 0$$P= \sum_{j=1}^{n}aj\frac{\partial}{\partial z_{j}}+a_{0}$$f\in O_{X,\mathit{0}}$annihilates the cohomology class , we have$\sigma$$P(f\sigma)$$=$$PF\sigma$$=$$(PF-FP)\sigma+FP\sigma$$=$.

$\sum_{j=1}^{n}a_{\mathrm{j}}\frac{\partial f}{\partial z_{j}}\sigma$

54
Since

$\sum_{j=1}^{n}a_{j}f_{j}\in I$,$P(f\sigma)=0$

holds. Thus, there exist at least 2elements
are linearly independent elements, we have$\sigma$and$f\sigma$in $Hm_{D_{X}}(D_{X}/Ann\leq 1,\mathcal{H}_{[O]}^{n}(O_{X}))$ . As$\sigma$and$f\sigma$.

$\dim Hom_{D_{X}}(D_{X}/Ann\leq 1,\mathcal{H}_{[O]}^{n}(O_{X}))\geq 2$$(\mathrm{i}\mathrm{i})\Rightarrow(\mathrm{i}\mathrm{i}\mathrm{i})$: By assumption, we have
$Hom_{D_{X}}(Dx/Ann\leq 1,\mathcal{H}_{[O]}^{n}(O_{X}))=Hom_{D_{X}}(D_{X}/Ann,\mathcal{H}_{[O]}^{n}(O_{X}))$

where $Ann=\{P\in D_{X}|P\sigma=0\}$ . Since $D_{X}/Ann$ is simple holonomic system, we have
$Hom_{D_{X}}Vx/Ann$$\mathcal{H}_{[O]}^{n}(O_{X}))=\mathrm{S}\mathrm{p}\mathrm{a}\mathrm{n}\{\sigma\}$.

$\square$

For the holomorphic function$f$with an isolated singularity at the origin\cdots ,$f_{n}\rangle=O_{X,O}(f_{1},$ \cdots ,$O_{X,O}\langle f, f_{1},$$f_{n}\rangle$$O$, suppose that

.

(3.2)

It is known in [2] there exists some holomorphic coordinate transformation which makes aquasihomoge
neous polynomial. Theorem 3.1 asserts that, it is possible to characterize the algebraic local cohomology
class attached to aquasihomoegeous singularity as the solution of the system of differential equations$f$$\sigma$of at most first order.

4The case of exceptional families of singularities of modality 1.
In this Section, we characterize cohomology classes attached to exceptional families of unimodal singularities. Functions having non-degenerate quasihomogeneous principal part of modality 1can be reduced tothree one parameter families of parabolic singularities and 14 polynomials generating exceptional families.
Since the parabolic singularities satisfy (3.2), our objects are the following 14 polynomials.
$2\mathrm{v}\mathrm{a}\mathrm{r}\mathrm{i}\mathrm{a}\mathrm{b}\mathrm{l}\mathrm{e}\mathrm{s}$

:
:
:
:
:
:
:
:

$E_{12}$$E_{13}$$E_{14}$$Z_{11}$$Z_{12}$$Z_{13}$$W_{12}$$W_{13}$$f(x,y)=x^{3}+y^{7}+axy^{5}$$3\mathrm{v}\mathrm{a}\mathrm{r}\mathrm{i}\mathrm{a}\mathrm{b}\mathrm{l}\mathrm{e}\mathrm{s}$$f(x,y)=x^{3}+xy^{5}+ay^{8}$$Q_{10}$$f(x,y)=x^{3}+y^{8}+axy^{6}$$Q_{11}$$f(x,y)=x^{3}y+y^{5}+axy^{4}$$Q_{12}$$f(x,y)=x^{3}y+xy^{4}+ay^{6}$$S_{11}$$f(x,y)=x^{3}y+y^{6}+axy^{5}$$S_{12}$$f(x,y)=x^{4}+y^{5}+ax^{2}y^{3}$$U_{12}$

:
:
:
:
:
:

$f(x, y, z)=x^{3}+y^{4}+yz^{2}+axy^{3}$$f(x,y,z)=x^{3}+y^{2}z+xz^{3}+az^{5}$$f(x,y, z)=x^{3}+y^{5}+yz^{2}+axy^{4}$$f(x,y,z)=x^{4}+y^{2}z+xz^{2}+ay^{2}x^{2}$$f(x,y,z)=x^{2}y+y^{2}z+xz^{3}+az^{5}$$f(x,y, z)=x^{3}+y^{3}+z^{4}+axyz^{2}$$f(x,y)=x^{4}+xy^{4}+ay^{6}$

These normal forms of quasihomogeneous singularities are given by V.I.Arnold ([1]).

4.1

The quasidegrees of cohomology classes$\mathrm{e}\mathrm{x}\mathrm{p}\mathrm{r}\mathrm{r}\mathrm{e}\mathrm{s}\mathrm{s}\mathrm{i}\mathrm{o}\mathrm{n}\mathrm{L}\mathrm{e}\mathrm{t}\alpha=(\alpha_{1}, \ldots,\alpha_{n})$be atyPe of quasihomogeneous singularities. Acohomology class$\eta=[\sum_{\mathrm{k}\in E}c_{\mathrm{k}}\frac{1}{z^{\mathrm{k}}}]$

where$\alpha$$\in \mathrm{Q}$and$z^{\mathrm{k}}=z\mathrm{f}^{1}\cdots$ $z_{n}^{k_{\mathrm{n}}}$with

Definition 4.1 A cohomology class$\mathrm{k}=$$[ \frac{1}{z^{\mathrm{k}}}]$$(k_{1}, \ldots, k_{n})$and $E$ is afinite subset of Nn.

has $degm-d$ if$(\alpha,\mathrm{k}\rangle=\alpha_{1}k_{1}+\cdots+\alpha_{n}k_{n}=d$$\eta\in\Sigma$

has an55
For acohomology class$[ \frac{1}{z^{\mathrm{k}}}]$

1n$\eta$$\eta=[\sum_{\mathrm{k}\in E_{\eta}}c_{\mathrm{k}}\frac{1}{z^{\mathrm{k}}}]$, we define its degree$d(\eta)$by the smallest degree of classes:$\mathrm{d}(\mathrm{n})=\min\{-\langle\alpha,\mathrm{k}\rangle |\mathrm{k} \in E_{\eta}\}$,the above expression of theof non-zero term
is aset of all exponents
where.
cohomology class . For both functions and cohomology classes, we denote its degree by
In the case of semiquasihomogeneous singularities, we have the following result.
$\mathrm{k}=$$E_{\eta}$$(k_{1}, \ldots, k_{n})$$\alpha\frac{1}{z^{\mathrm{k}}}.\mathrm{n}$$d(\cdot)$$\eta$

Proposition 4.1 Let be a semiquasihomogeneous function. For any basis monomial $mj$ of the vector
which satisfies following two, there exists a cohomology class in the vector space
space
conditions :, where is the delta function with support at the origin.
(i)$f$$\Sigma$$O_{X,\mathit{0}}/I$$\eta$$m_{j}\eta=\delta$$\delta$(ii) $d( \eta)=-\sum_{j=1}^{n}\alpha_{j}-d(m_{j})$ .
Furthermore, we have the following proposition.

Proposition 4.2 Let be a semiquasihomogeneous function. A necessary and sufficient condition foristo be a generator of 1overcohomology class$f$$\sigma\in\Sigma$$a$$Ox,\mathit{0}$$d( \sigma)=-nd(f)+\sum_{j=1}^{n}\alpha_{\mathrm{j}}$.

Cohomology classes attached to exceptional singularities of modality 1.

4.2

Recall that, for anon-quasihomogeneous function , we have
$f$

(4.1)

$O_{X,\mathrm{O}}\langle f_{1}, \ldots, f_{n}\rangle\neq O_{X,O}\langle f, f_{1}, \ldots, f_{n}\rangle$

and thus.

$\dim Hom_{D_{X}}$ $(Dx/Ann\leq 1,\mathcal{H}_{[O]}^{n}(Ox))\geq 2$

We examine the solution space $Homv_{X}$ $(Dx/Ann\leq 1,\mathcal{H}_{[O]}^{n}(Ox))$ of the holonomic system $Dx/Ann\leq 1$
attached to exceptional singularities of modality 1.
. That is, we have theWe verify that 7{ is spanned by 1and the modulo class of $f(z)$ in
following proposition.
Proposition 4.3 For a function defining an exceptional singularity of modality 1, we have
$Ox,\mathit{0}/I$

$f$

$7\#$

$=\mathrm{S}\mathrm{p}\mathrm{a}\mathrm{n}\{1,$fmod I}.

Proposition 4.3 is proved by direct computations for each normal form of an exceptional family of$(j=1, \ldots, n)$ , we havemod I where
singularities of modality 1. Since$f\mathrm{m}\mathrm{o}\mathrm{d} I=c_{\mathrm{O}}j_{F}(z)$$z_{\mathrm{j}}f\in I$

andis anon-zero constant. Thus we have the following theorem.

$Jf(z)$ is the Jacobian$\frac{\partial(f_{1},\ldots,f_{n})}{\partial(z_{1},\ldots,z_{n})}$Theorem 4.1 Let

be a function defining an exceptional singularity of modality 1. Then, we have$f$$c_{\mathrm{O}}$,$Hom_{D_{X}}$ $(D_{X}/Ann\leq 1,\mathcal{H}_{[O]}^{n}(O\mathrm{x}))=\mathrm{S}\mathrm{p}\mathrm{a}\mathrm{n}\{\sigma,\delta\}$

where$\delta$

is the delta function with support at the origin$O$.

To give effects of computations, we introduce the following vector spaces :,$L= \{P=\sum_{j=1}^{n}a_{j}(z)\frac{\partial}{\partial z_{\mathrm{j}}}+a_{0}(z)|P\sigma=0, a_{\mathrm{j}}(z)\in O_{X,O}/I, j=0, \ldots,n\}$,$V= \{\mathrm{v}=\sum_{j=1}^{n}a_{j}(z)\frac{\partial}{\partial z_{\mathrm{j}}}|\mathrm{v}g\in I^{\forall},g\in I, a_{j}(z)\in Ox,\mathit{0}/I, j=1, \ldots,n\}$$H=\{h\in Ox,\mathit{0}/I|\mathrm{v}h=0, \forall \mathrm{v}\in V\}$.

56
Lemma 4.1 We have the isomorphism between

$L$

and

$V$

:

.

$L\underline{\simeq}V$

Proof. For any. Cl$\mathrm{v}\in V$, there exist$h\in Ox,o\mathrm{s}.\mathrm{t}.$,$\mathrm{v}\sigma=\mathrm{h}\mathrm{a}$. By putting$a_{0}=-h\mathrm{m}\mathrm{o}\mathrm{d} I$, we have$(\mathrm{v}+a_{0})\sigma=0$

4.3

Example :singularity.

$E_{12}$

The quasihomogeneous part of the function $f=x^{3}+y^{7}+axy^{5}$ is of type $(7, 3)$ of degree 21. The
partial derivatives of $f$ with respect to the variables and are $f_{x}=3x^{2}+ay^{5}$ and $f_{y}=7y^{6}+5axy^{4}$ ,respectively. We use the lexicographic order with $x\succ y$ in computations. The standard base of the ideal$x$$I=( \frac{\partial f}{\partial x},$$\frac{\partial f}{\partial y}\rangle_{\mathit{0}}$$y$in $O_{X,O}$ is$\{y^{8},7y^{6}+5ay^{4}x,3x^{2}+ay^{5}\}$.

Basis monomials of the local ring $O_{X,O}/I$ is given by
, x,
6, 7,

1, y,
0, 3,

$y^{2}$$y^{3}$

, yx,

9,

10,

$7y^{2}x\partial_{l}-2ayx\partial_{y}$,$y^{2}x$12,13,$-3\mathrm{O}\mathrm{a}\mathrm{x}0\mathrm{y}+65a^{2}y^{4}\partial_{l}$$252\mathrm{y}\mathrm{x}0,$$y^{4}\partial_{y}$,$y^{4}$$y^{3}x\partial_{x}$,$y^{2}x\partial_{y}$

,

,

,

,

,

$y^{3}x$16,,$y^{4}x$,$y^{5}x$2219,$63y^{2}\partial_{y}+15ax\partial_{y}+25a^{2}y^{4}\partial_{l}$$7y^{3}\partial_{y}+5ayx\partial_{y}$$y^{5}\partial_{y}$,15,$y^{5}$$y^{4}x\partial_{*}$,,$y^{3}x\partial_{y}$$6yx\partial_{y}-5ay^{5}\partial_{\mathrm{g}}$,$y^{5}x\partial_{l}$,$y^{4}x\partial_{y}$

,

,

,

$y^{5}x\partial_{y}$The solution space of the simultaneous homogeneous equation $\mathrm{v}h(x,y)=0$ (Vv $\in V$ ) is
$H=\mathrm{S}\mathrm{p}\mathrm{a}\mathrm{n}\{1,y^{5}x\}$

Note that $y^{5}x=f$ mod$I$.

. Since $xf,yf\in I$ , we have
$Hm\mathrm{p}_{X}$ $(D_{X}/Ann\leq 1,fl_{[O]}^{n}(O_{X}))=\mathrm{S}\mathrm{p}\mathrm{a}\mathrm{n}\{\sigma,\delta\}$

where$\delta$,

is the delta function with support at the origin.

4.4 Computations for normal forms
In this section, we give results of computations for normal forms of singularities of modality 1listed
before. To compute the solution space $Homv_{\mathrm{x}}(Dx/Ann\leq 1,?t_{[O]}^{n}(O_{X}))$ , we give$\bullet$partial derivatives$\bullet$the standard base of the ideal I of partial derivatives of $f(z)$ at the origin,$\bullet$.
$\bullet$Basis monomials

basis
basis

$\sigma_{1}$$\mathrm{v}_{1}$,,$\ldots$$\ldots$,,$\sigma_{\mu}$$\mathrm{v}_{N}$$f_{z_{j}}$$m_{1}$,of the function $f(z)$ ,$\ldots$,$m_{\mu}$of $O_{X}/I$ and its degree,of the vector spaceof the vector space$\Sigma$$V$and its degree,
and its degree,57
and $H$ as the solution space of the simultaneous homogeneous equations $\mathrm{v}h(z)=0$ for every. The
number below the.basis of, , and is its degree. Here, each has weightand eachhas$\mathrm{v}\in V$$\mathrm{w}\mathrm{e}\mathrm{i}\mathrm{g}\mathrm{h}\mathrm{t}-\alpha_{\mathrm{j}}$$V$$\Sigma$$\mathrm{O}\mathrm{x},0/\mathrm{I}$$\partial_{\mathrm{j}}$$z_{j}$$\alpha_{\mathrm{j}}$.

We give basisof which satisfies Proposition 4.1 for every basis monomial $m_{\mu-j+1}$ of,$\mu=\dimO_{X,\mathit{0}}/I$
whereis the Milnor number. That is,is the dual base of the monomial baseof $O_{X,O}/I$ . The cohomology classgenerates over $O_{X,O}$ . Note that in expressions of
the basis of , we find the basis of the set of local cohomology classes annihilated by partial derivativesof quasihomogeneous part of the function if we substitute $a=0$ .
We use the standard basis in computations with respect to the lexicographic order withorofwhere :is the weight of the variable :(resp. , ). Therefore, the,monomial basis of the local ring $O_{X,O}/(f_{1}$ ,,of Z12,
andused in this paper are
different from that in [1],$\Sigma$$\mathrm{O}\mathrm{x},0/\mathrm{I}$$\sigma_{j}$$\{\sigma_{1}, \ldots, \sigma_{\mu}\}$$\{m_{\mu}, \ldots, m_{1}\}$$\Sigma$$\sigma_{1}$$\Sigma$$f$$Z:\succ z_{\mathrm{j}}$$\alpha_{i}\geq\alpha j\geq\alpha_{k}$$z_{i}\succ z_{j}\succ z_{k}$$\ldots$

4.4.1$E_{12}$$Q_{10}$$f_{n}\rangle$

$k$

$j$

$Z$

$\alpha$$S_{11}$$S_{12}$: $x^{3}+y^{7}+axy^{5}$$f=x^{3}+y^{7}+axy^{5}$ (of type (7, 3) of degree 21)
Partial derivatives $:.f_{x}=3x^{2}+ay^{5}$ , $f_{y}=7y^{6}+5axy^{4}$
The standard base of: $\{y^{8},7y^{6}+5ay^{4}x, 3x^{2}+ay^{5}\}$
Basis of the local ringand its degrees :$I=\langle f_{x}, f_{y}\rangle \mathit{0}$$\mathrm{O}\mathrm{x}\mathrm{f}\mathrm{o}/\mathrm{I}$$m_{1}$$m_{3}$$m_{4}$$m_{5}$$m_{6}$$m_{7}$$m_{8}$$m_{9}$$m_{10}$$m_{11}$$m_{12}$$y^{2}$$x$$y^{3}$$yx$$y^{4}$$y^{2}x$$y^{5}$$y^{3}x$$y^{4}x$$y^{5}x$0367910

12

13

15

16

19

22

$m_{2}$

1$y$Basis of I:$\sigma_{1}=[\frac{1}{y^{6}x^{2}}+a(-\frac{5}{7}\frac{1}{y^{8}x}-\frac{1}{3}\frac{1}{yx^{4}})+\frac{5}{21}a^{2}\frac{1}{y^{3}x^{3}}]$,$\sigma_{3}=[\frac{1}{y^{4}x^{2}}]$$\sigma_{4}=[\frac{1}{y^{6}x}-\frac{1}{3}a\frac{1}{yx^{3}}]$$\sigma_{9}=[\frac{1}{yx^{2}}]$,,$\sigma_{5}=[\frac{1}{y^{3}x^{2}}]$,$\sigma_{10}=[\frac{1}{y^{3}x}]$,$\sigma_{2}=[\frac{1}{y^{5}x^{2}}-\frac{5}{7}a\frac{1}{y^{7}x})+\frac{21}{5}a^{2}\frac{1}{y^{2}x^{3}}]$,$\sigma_{6}=[\frac{1}{y^{5}x}]$$\sigma_{11}=[\frac{1}{y^{2}x}]$,,$\sigma_{7}=[\frac{1}{y^{2}x^{2}}]$,$\sigma_{8}=[\frac{1}{y^{4}x}]$$\sigma_{12}=[\frac{1}{yx}]$

Degrees :$\sigma_{2}$$\sigma_{1}$Basis of$\sigma_{5}$-29 -26 -25

-32
$V$

$\sigma_{4}$$\sigma_{3}$$\sigma_{6}$$\sigma_{8}$$\sigma_{7}$$\sigma_{10}$$\sigma_{9}$$\sigma_{12}$$\sigma_{11}$-23 -22 -20 -19 -17 -16 -13 -10:$\mathrm{v}_{1}=252yx\partial_{x}-30ax\partial_{y}+65a^{2}y^{4}\partial_{x}$$\mathrm{v}_{3}=7y^{2}x\partial_{x}-2ayx\partial_{y}$$\mathrm{v}_{6}=y^{4}\partial_{y}$,$\mathrm{v}_{7}=y^{3}x\partial_{\mathrm{g}}$

,

,

,

$\mathrm{v}_{2}=63y^{2}\partial_{y}+15ax\partial_{y}+25a^{2}y^{4}\partial_{x}$$\mathrm{v}_{4}=+7y^{3}\partial_{y}+5ayx\partial_{y}$$\mathrm{v}_{8}=y^{2}x\partial_{y}$$\mathrm{v}_{12}=y^{5}x\partial_{x}$

,

, Vg

$=y^{5}\partial_{y}$$\mathrm{v}_{13}=y^{4}x\partial_{y}$

,

,

,

$\mathrm{v}_{5}=6yx\partial_{y}-5ay^{5}\partial_{x}$$\mathrm{v}_{10}=y^{4}x\partial_{x}$,Degrees :$\mathrm{V}\mathrm{i}$$\mathrm{v}_{2}$V3$\mathrm{v}_{4}$$\mathrm{v}_{5}$$\mathrm{v}_{6}$$\mathrm{v}_{7}$$\mathrm{v}_{8}$$\mathrm{V}\mathfrak{g}$336679910 12
Solution space $H$ :.

$H=\mathrm{S}\mathrm{p}\mathrm{a}\mathrm{n}\{1,y^{5}x\}$,,$\mathrm{v}_{11}=y^{3}x\partial_{y}$$\mathrm{v}_{14}=y^{5}x\partial_{y}$$\mathrm{v}_{10}$$\mathrm{v}_{11}$$\mathrm{v}_{12}$$\mathrm{v}_{13}$$\mathrm{v}_{14}$

12

13

15

16

19

,

,

,

58
4.4.2

: $x^{3}+xy^{5}+ay^{8}$$E_{13}$$f=x^{3}+xy^{5}+ay^{8}$ (of type (5, 2) of degree 15)

Partial derivatives : $f_{ax}=3x^{2}+y^{5}$ , $f_{y}=5xy^{4}+8ay^{7}$: $\{y^{9},5y^{4}x+8ay^{7},3x^{2}+y^{5}\}$The standard base of $I=(f_{l},$and its degrees :Basis of the local ring$f_{y}\rangle$$0$$\mathrm{O}\mathrm{x},0/\mathrm{I}$$m_{2}$$m_{1}$1$y$$m_{3}$$m_{4}$$m_{5}$$m_{6}$$m_{7}$$m_{8}$$m_{9}$$m_{10}$$m_{11}$$m_{12}$$m_{13}$$y^{2}$$x$$y^{3}$$yx$$y^{4}$$y^{2}x$$y^{5}$$y^{3}x$$y^{6}$$y^{7}$$y^{8}$

11

12

14

16

0245678910
Basis of:$\Sigma$$\sigma_{1}=[\frac{1}{y^{9}x}-\frac{1}{3}\frac{1}{y^{4}x^{3}}+a(-\frac{8}{5}\frac{1}{y^{6}x^{2}}+\frac{8}{15}\frac{1}{yx^{4}})]$$\sigma_{3}=[\frac{1}{y^{7}x}-\frac{1}{3}\frac{1}{y^{2}x^{3}}]$$\sigma_{8}=[\frac{1}{y^{2}x^{2}}]$,,$\sigma_{4}=[\frac{1}{y^{4}x^{2}}]$$\sigma_{9}=[\frac{1}{y^{4}x}]$

,

,

,

$\sigma_{2}=[\frac{1}{y^{8}x}-\frac{1}{3}\frac{1}{y^{3}x^{3}}-\frac{8}{5}a\frac{1}{y^{5}x^{2}}]$,$\sigma_{5}=[\frac{1}{y^{6}x}-\frac{1}{3}\frac{1}{yx^{3}}]$$\sigma_{10}=[\frac{1}{yx^{2}}]$,$\sigma_{11}=[\frac{1}{y^{3}x}]$,,$\sigma_{6}=[\frac{1}{y^{3}x^{2}}]$$\sigma_{12}=[\frac{1}{y^{2}x}]$,,,$\sigma_{7}=[\frac{1}{y^{5}x}]$$\sigma_{13}=[\frac{1}{yx}]$

Degrees :$V$$\sigma_{8}$$\sigma_{11}$$\sigma_{10}$$\sigma_{9}$-19 -18 -17 -16 -15 -14 -13 -12 -11-23 -21
Basis of$\sigma_{7}$$\sigma_{6}$$\sigma_{5}$$\sigma_{4}$$\sigma_{3}$$\sigma_{2}$$\sigma_{1}$$\sigma_{12}$$\sigma_{13}$

-9

-7

:

,

$\mathrm{v}_{1}=125\mathrm{y}\mathrm{x}\mathrm{a}\mathrm{x}+50y^{2}\partial_{y}-40ay^{4}\partial_{l}+192a^{2}y^{2}x\partial_{l}$$\mathrm{v}_{2}=20y^{4}\partial_{l}+15x\partial_{y}-15\infty y^{2}x\partial_{x}$

,

’

,

$\mathrm{v}0=5y^{3}x\partial_{ae}+2y^{4}\partial_{y},\mathrm{v}_{7}=y^{2}x\partial_{l},\mathrm{v}_{8}=y^{6}\partial_{l},\mathrm{v}_{9}=y^{5}\partial_{y},\mathrm{v}_{10}=y^{3}x\partial_{y}\mathrm{v}_{3}=5y^{2}x\partial_{ae}+2y^{3}\partial_{y},\mathrm{v}_{4}=yx\partial_{y}-4ay^{3}x\partial_{x},\mathrm{v}_{5}=5y^{5}\partial_{l}-24ay^{3}x\partial_{l}\mathrm{v}_{11}=y^{7}\partial_{l},\mathrm{v}_{12}=y^{6}\partial_{y},\mathrm{v}_{13}=y^{8}\partial_{l},\mathrm{v}_{14}=y^{7}\partial_{y},\mathrm{v}_{15}=y^{8}\partial_{y}$Degrees :V3$\mathrm{v}_{2}$$\mathrm{v}_{1}$V4$\mathrm{V}8$$\mathrm{V}7$$\mathrm{v}_{6}$$\mathrm{v}_{5}$$\mathrm{v}_{10}$$\mathrm{V}\mathfrak{g}$$\mathrm{v}_{12}$$\mathrm{v}_{11}$2345567789910
Solution space H : H
4.4.3$E_{14}$$\mathrm{V}13$$\mathrm{v}_{14}$$\mathrm{v}_{15}$

11

12

14

$=\mathrm{S}\mathrm{p}\mathrm{a}\mathrm{n}\{1,y^{8}\}$: $x^{3}+y^{8}+axy^{6}$$f=x^{3}+y^{8}+axy^{6}$ (of type (8, 3) of degree 24)

Partial derivatives : $f_{l}=3x^{2}+ay^{6}$ , $f_{y}=8y^{7}+6axy^{5}$: $\{y^{9},4y^{7}+3ay^{5}x,3x^{2}+ay^{6}\}$The standard base of$O_{X,O}/I$
and its degrees :Basis of the local ring
$I=(f_{l},f_{y})\mathit{0}$$m_{3}$$m_{4}$$m_{5}$$m_{6}$$m_{7}$$m_{8}$$m\mathrm{g}$$m_{10}$$m_{11}$$m_{12}$$m_{13}$$m_{14}$$y^{2}$$x$$y^{3}$$yx$$y^{4}$$y^{2}x$$y^{5}$$y^{3}x$$y^{6}$$y^{4}x$$y^{5}x$$y^{6}x$0368911

12

14

15

17

18

20

23

26

$m_{1}$$m_{2}$

1

Basis of$\Sigma$$y$:$\sigma_{1}=[\frac{1}{y^{7}x^{2}}+a(-\frac{3}{4}\frac{1}{y^{9}x}-\frac{1}{3}\frac{1}{yx^{4}})+\frac{1}{4}a^{2}\frac{1}{y^{3}x^{3}})]$$\sigma_{3}=[\frac{1}{y^{5}x^{2}}]$,$\sigma_{4}=[\frac{1}{y^{7}x}-\frac{1}{3}a\frac{1}{yx^{3}}]$$\sigma_{9}=[\frac{1}{y^{2}x^{2}}]$,$\sigma_{10}=[\frac{1}{y^{4}x}]$,,$\sigma_{5}=[\frac{1}{y^{4}x^{2}}]$$\sigma_{11}=[\frac{1}{yx^{2}}]$

,

,

,

$\sigma_{2}=[\frac{1}{y^{6}x^{2}}-\frac{3}{4}a\frac{1}{y^{8}x}+\frac{1}{4}a^{2}\frac{1}{y^{2}x^{3}}]$$\sigma_{6}=[\frac{1}{y^{6}x}]$$\sigma_{12}=[\frac{1}{y^{3}x}]$,,$\sigma_{7}=[\frac{1}{y^{3}x^{2}}]$$\sigma_{13}=[\frac{1}{y^{2}x}]$

,

,

,

$\sigma_{8}=[\frac{1}{y^{5_{\tau}}}$$\sigma_{14}=[\frac{1}{yx}]$

59
04

03

(

$\mathrm{r}_{2}$$\mathrm{c}\mathrm{r}_{1}$

010

09

08

”7

”6

$\mathit{0}_{\mathit{5}}$

0gg

014

013

$\mathit{0}_{\mathit{1}\mathit{2}}$-37 -34 -31 -29 -28 -26 -25 -23 -22 -20 -19 -17 -14 -11
Basis of$V$:$\mathrm{v}_{1}=28yx\partial_{\mathrm{r}}-3ax\partial_{y}-4a^{2}y^{5}\partial_{x}$$\mathrm{v}_{3}=4y^{2}x\partial_{ae}-ayx\partial_{y}$,,$\mathrm{v}_{2}=28y^{2}\partial_{y}+6ax\partial_{y}+15a^{2}y^{5}\partial_{l}$$\mathrm{v}_{4}=4y^{3}\partial_{y}+3ayx\partial_{y}$,$\mathrm{v}_{5}=yx\partial_{y}-ay^{6}\partial_{l}$,,,$\mathrm{v}_{12}=y^{6}\partial_{y},\mathrm{v}_{13}=y^{5}x\partial_{x},\mathrm{v}_{14}=y^{4}x\partial_{y},\mathrm{v}_{15}=y^{6}x\partial_{x},\mathrm{v}_{16}=y^{5}x\partial_{y},\mathrm{v}_{17}=y^{6}x\partial_{y}\mathrm{v}_{6}=y^{4}\partial_{y},\mathrm{v}_{7}=y^{3}x\partial_{x},\mathrm{v}_{8}=y^{2}x\partial_{y},\mathrm{v}_{9}=y^{5}\partial_{y},\mathrm{v}_{10}=y^{4}x\partial_{x},\mathrm{v}_{11}=y^{3}x\partial_{y}$Degrees :$\mathrm{v}_{1}$$\mathrm{v}_{2}$

v4

V3

vg

$\mathrm{v}_{8}$$\mathrm{v}_{7}$$\mathrm{v}_{6}$$\mathrm{v}_{5}$$\mathrm{v}_{10}$$\mathrm{v}_{11}$$\mathrm{v}_{12}$$\mathrm{v}_{13}$$\mathrm{v}_{14}$$\mathrm{v}_{15}$$\mathrm{v}_{16}$$\mathrm{v}_{17}$

12

14

15

15

17

18

20

23

336689941 12
Solution space H : H
4.4.4$Z_{11}$$=\mathrm{S}\mathrm{p}\mathrm{a}\mathrm{n}\{1,y^{6}x\}$: $x^{3}y+y^{5}+axy^{4}$$f=x^{3}y+y^{5}+axy^{4}$ (of type (4, 3) of degree 15)

Partial derivatives : $f_{x}=3x^{2}y+ay^{4}$ , $f_{y}=x^{3}+5y^{4}+4axy^{3}$: $\{y^{6},15y^{5}+11ay^{4}x,3yx^{2}+ay^{4},x^{3}+5y^{4}+4ay^{3}x\}$
The standard base ofdegrees :itsandBasis of the local ring
$I=\langle f_{x}, f_{y}\rangle \mathit{0}$$Ox,\mathit{0}/I$$m_{1}$$m_{2}$$m_{3}$$m_{4}$$m_{5}$$m_{6}$$m_{7}$$m_{8}$$m_{9}$$m_{10}$$m_{11}$$y$$x$$y^{2}$$yx$$x^{2}$$y^{3}$$y^{2}x$$y^{4}$$y^{3}x$$y^{4}x$

12

13

16

1

034678910
Basis of I:$\sigma_{1}=[\frac{1}{y^{5}x^{2}}-5\frac{1}{yx^{5}}+a(-\frac{11}{15}\frac{1}{y^{6}x}-\frac{1}{3}\frac{1}{y^{2}x^{4}})+\frac{11}{45}a^{2}\frac{1}{y^{3}x^{3}}]$$\sigma_{3}=[\frac{1}{y^{5}x}-5\frac{1}{yx^{4}}-\frac{1}{3}a\frac{1}{y^{2}x^{3}}]$$\sigma_{8}=[\frac{1}{y^{3}x}]$$-20\sigma_{2}$$-23\sigma_{1}$,,$\sigma_{4}=[\frac{1}{y^{3}x^{2}}]$$-17\sigma_{4}$$-19\sigma_{3}$,$\sigma_{9}=[\frac{1}{yx^{2}}]$$-16\sigma_{5}$,,$\sigma_{2}=[\frac{1}{y^{4}x^{2}}-\frac{4}{5}a\frac{1}{y^{5}x}+\frac{4}{15}a^{2}\frac{1}{y^{2}x^{3}}]$,$\sigma_{5}=[\frac{1}{y^{4}x}]$$\sigma_{10}=[\frac{1}{y^{2}x}]$$-15\sigma_{6}$$-14\sigma_{7}$,$\sigma_{6}=[\frac{1}{yx^{3}}]$,$\sigma_{7}=[\frac{1}{y^{2}x^{2}}]$$\sigma_{11}=[\frac{1}{yx}]$$-11\sigma_{9}$$-13\sigma_{8}$$-10\sigma_{10}$$\sigma_{11}-7$

Basis of $V$ :,$\mathrm{v}_{2}=15y^{2}\partial_{y}+a(108x^{2}\partial_{x}+83yx\partial_{y})$$\mathrm{v}_{1}=15yx\partial_{x}-a(61x^{2}\partial_{x}+48yx\partial_{y})$$\mathrm{v}_{3}=60x^{2}\partial_{x}+45yx\partial_{y}+ax^{2}\partial_{y}$V5$=y^{3}\partial_{y}$,$\mathrm{v}_{6}=y^{2}x\partial_{x}$$\mathrm{v}_{10}=y^{3}x\partial_{x}$

,

,

,

$\mathrm{v}_{7}=y^{2}x\partial_{y}$$\mathrm{v}_{11}=y^{3}x\partial_{y}$,$\mathrm{v}_{4}=5y^{3}\partial_{x}+2x^{2}\partial_{y}$,$\mathrm{v}_{8}=y^{4}\partial_{x}$$\mathrm{v}_{12}=y^{4}x\partial_{x}$

,

,

,

$\mathrm{v}_{9}=y^{4}\partial_{y}$,$\mathrm{v}_{13}=y^{4}x\partial_{y}$Degrees :$\mathrm{v}_{1}$v2$\mathrm{v}_{3}$v4$\mathrm{v}_{5}$$\mathrm{v}_{6}$$\mathrm{v}\tau$$\mathrm{v}_{8}$$\mathrm{v}_{9}$$\mathrm{v}_{10}$$\mathrm{v}_{11}$334566789910
Solution space $H$ :$H=\mathrm{S}\mathrm{p}\mathrm{a}\mathrm{n}\{1,y^{4}x\}$$\mathrm{v}_{12}$$\mathrm{v}_{13}$

12

13

,

,

,

60
4.4.5

$Z_{12}$: $x^{3}y+xy^{4}+ay^{6}$$f=x^{3}y+xy^{4}+ay^{6}$ (of tyPe (3, 2) of degree 11)
Partial derivatives : $f_{x}=3x^{2}y+y^{4}$ , $f_{y}=x^{3}+4xy^{3}+6ay^{5}$
The standard base of $I=(f_{l},$:$f_{y}\rangle_{\mathit{0}}$$\{y^{7},33y^{4}x-7ay^{6},3yx^{2}+y^{4}+2ay^{3}x,33x^{3}+132y^{3}x-33ay^{5}-14a^{3}y^{6}\}$

Basis of the local ring$m_{1}$

1
0
Basis of$\Sigma$and its degrees :$\mathrm{O}\mathrm{x},0/\mathrm{I}$$m_{2}$$m_{3}$$m_{4}$$m_{5}$$m_{6}$$m_{7}$$m_{8}$$m_{9}$$m_{10}$$m_{11}$$m_{12}$$y$$x$$y^{2}$$yx$$x^{2}$$y^{3}$$y^{2}x$$y^{4}$$y^{3}x$$y^{5}$$y^{6}$

2

3

4

5

6

6

7

8

9

10

12

:

,

$\sigma_{1}=[\frac{1}{y^{7}x}-\frac{1}{3}\frac{1}{y^{4}x^{3}}+\frac{4}{3}\frac{1}{yx^{5}}+a(\frac{6}{11}\frac{1}{y^{2}x^{4}}-\frac{18}{11}\frac{1}{y^{5}x^{2}})]$$\sigma_{3}=[\frac{1}{y^{4}x^{2}}-4\frac{1}{yx^{4}}]$$\sigma_{8}=[\frac{1}{y^{2}x^{2}}]$,$\sigma_{4}=[\frac{1}{y^{5}x}-\frac{1}{3}\frac{1}{y^{2}x^{3}}]$,,$\sigma_{9}=[\frac{1}{y^{3}x}]$,$\sigma_{2}=[\frac{1}{y^{6}x}-\frac{1}{3}\frac{1}{y^{3}x^{3}}-\frac{3}{2}a\frac{1}{y^{4}x^{2}}]$$\sigma_{5}=[\frac{1}{y^{3}x^{2}}]$$\sigma_{10}=[\frac{1}{yx^{2}}]$,,$\sigma_{6}=[\frac{1}{y^{4}x}]$$\sigma_{11}=[\frac{1}{y^{2}x}]$,$\sigma 12$,$\sigma_{7}=[\frac{1}{yx^{3}}]$,,$=[ \frac{1}{yx}]$

Degrees :$\sigma_{1}$Basis of$V$$\sigma_{2}$$\sigma_{3}$$\sigma_{4}$$\sigma_{5}$$\sigma_{6}$$\sigma_{7}$$\sigma_{8}$$\sigma_{10}$$\sigma_{11}$$\sigma_{12}$-17 -15 -14 -13 -12 -11 -11 -10 -9 -8-7-5$\sigma_{9}$:,$\mathrm{v}_{1}=1936y^{2}\partial_{l}-13\infty 8ayx\partial_{l}+2673a^{2}x^{2}\partial_{l}+4374a^{3}y^{2}x\partial_{l}+1452\mathrm{x}0\mathrm{y}$$\mathrm{v}_{2}=132yx\partial_{x}+88y^{2}\partial_{y}-99ax^{2}\partial_{l}-162a^{2}y^{2}x\partial_{ae}$$\mathrm{v}_{4}=4y^{3}\partial_{l}+3yx\partial_{y}-27ay^{2}x\partial_{l}$$\mathrm{v}_{7}=y^{2}x\partial_{y}$,$\mathrm{v}_{8}=y^{4}\partial_{x}$$\mathrm{v}_{12}=y^{5}\partial_{x}$

,

,

,

,

$\mathrm{v}_{3}=3x^{2}\partial_{l}+2yx\partial_{y}$$\mathrm{v}_{5}=3y^{2}x\partial_{l}+2y^{3}\partial_{y}$$\mathrm{v}_{9}=y^{4}\partial_{y}$$\mathrm{v}_{13}=y^{5}\partial_{y}$,,$\mathrm{v}_{10}=y^{3}x\partial_{x}$$\mathrm{v}_{14}=y^{6}\partial_{ae}$

,

,

,

$\mathrm{v}_{6}=8y^{3}\partial_{y}-9x^{2}\partial_{y}$,$\mathrm{v}_{11}=y^{3}x\partial_{y}$,,$\mathrm{v}_{15}=y^{6}\partial_{y}$Degrees :$\mathrm{v}_{1}$$\mathrm{v}_{2}$V3123
Solution space $H$ :4.4.6$Z_{13}$V5$\mathrm{v}_{4}$3V7$\mathrm{v}_{6}$

4

4

5

$\mathrm{v}_{8}$

Vg

5

6

$\mathrm{v}_{10}$$\mathrm{v}_{11}$$\mathrm{v}_{12}$$\mathrm{v}_{13}$$\mathrm{v}_{14}$$\mathrm{v}_{15}$

6

7

7

8

9

10

$H=\mathrm{S}\mathrm{p}\mathrm{a}\mathrm{n}\{1,y^{6}\}$: $x^{3}y+y^{6}+axy^{5}$$f=x^{3}y+y^{6}+axy^{5}$ (of tyPe (5, 3) of degree 18)
Partial derivatives :, $f_{y}=x^{3}+6y^{5}+5axy^{4}$$I=(f_{l},$The standard base of: $\{y^{7},9y^{6}+7ay^{5}x,3yx^{2}+\mathrm{a}\mathrm{y}5, x^{3}+6y^{5}+5ay^{4}x\}$
Basis of the local ringand its degrees:$f_{\mathrm{g}}=3x^{2}y+ay^{5}$$f_{y}\rangle_{\mathit{0}}$$\mathrm{O}\mathrm{x},0/\mathrm{I}$$m_{1}$

10$m_{2}$$m_{3}$$m_{4}$$m_{5}$$m_{6}$$m_{7}$$m_{8}$$m_{9}$$m_{10}$$m_{11}$$m_{12}$$m_{13}$$y$$x$$y^{2}$$yx$$y^{3}$$x^{2}$$y^{2}x$$y^{4}$$y^{3}x$$y^{5}$$y^{4}x$$y^{5}x$

3

5

6

8

9

10

11

12

14

15

17

20

Basis of I:,$\sigma_{1}=[\frac{1}{y^{6}x^{2}}-6\frac{1}{yx^{5}}+a(-\frac{7}{9}\frac{1}{y^{7}x}+\frac{7}{27}\frac{1}{y^{3}x^{3}}-\frac{1}{3}\frac{1}{y^{2}x^{4}})]$$\sigma_{3}=[\frac{1}{y^{5}x^{2}}-5a\frac{1}{yx^{4}}]$,$\sigma_{4}=[\frac{1}{y^{4}x^{2}}]$,$\sigma_{5}=[\frac{1}{y^{5}x}]$,$\sigma_{2}=[\frac{1}{y^{5}x^{2}}-\frac{5}{6}a\frac{1}{y^{6}x}+\frac{5}{18}a^{2}\frac{1}{y^{2}x^{3}}]$$\sigma_{6}=[\frac{1}{y^{3}x^{2}}]$,$\sigma_{7}=[\frac{1}{yx^{3}}]$,$\sigma_{8}=[\frac{1}{y^{4}x}]$61,$\sigma_{9}=[\frac{1}{y^{2}x^{2}}]$$\sigma_{10}=[\frac{1}{y^{3}x}]$,,$\sigma_{11}=[\frac{1}{yx^{2}}]$$\sigma_{12}=[\frac{1}{y^{2}x}]$,$\sigma_{13}=[\frac{1}{yx}]$

Degrees :$\sigma_{1}$$\sigma_{2}$$\sigma_{3}$$\sigma_{4}$$\sigma_{6}$$\sigma_{5}$$\sigma_{7}$$\sigma_{8}$$\sigma_{10}$$\sigma_{9}$$\sigma_{11}$$\sigma_{12}$$\sigma_{13}$-28 -25 -23 -22 -20 -19 -18 -17 -16 -14 -13 -11-8

Basis of V :$\mathrm{v}_{1}=9yx\partial_{l}-a(23x^{2}\partial_{x}+15yx\partial_{y})$$\mathrm{v}_{3}=45x^{2}\partial_{x}+27yx\partial_{y}+ax^{2}\partial_{y}$$\mathrm{v}_{8}=y^{4}\partial_{y}$,$\mathrm{v}_{13}=y^{4}x\partial_{l}$,$\mathrm{v}_{7}=y^{2}x\partial_{y}$

,

,

,

$\mathrm{v}_{2}=9y^{2}\partial_{y}+a(60x^{2}\partial_{x}+37yx\partial_{y})$,$\mathrm{v}_{4}=y^{3}\partial_{y}$$\mathrm{v}_{9}=y^{3}x\partial_{ae}$,$\mathrm{v}_{5}=y^{2}x\partial_{x}$,$\mathrm{v}_{10}=y^{5}\partial_{l}$$\mathrm{v}_{14}=y^{4}x\partial_{y}$,,$\mathrm{v}_{6}=3y^{4}\partial_{x}+x^{2}\partial_{y}$$\mathrm{v}_{11}=y^{3}x\partial_{y}$,$\mathrm{v}_{15}=y^{5}x\partial_{ax}$,,$\mathrm{v}_{12}=y^{5}\partial_{y}$,,$\mathrm{v}_{16}=y^{5}x\partial_{y}$Degrees :v2$\mathrm{v}_{1}$v4$\mathrm{v}_{3}$$\mathrm{v}_{6}$$\mathrm{v}_{5}$$\mathrm{v}_{7}$$\mathrm{v}_{8}$$\mathrm{v}_{10}$$\mathrm{v}_{9}$$\mathrm{v}_{11}$33566789910
Solution space H : H
4.4.7$W_{12}$11$\mathrm{v}_{12}$$\mathrm{v}_{13}$$\mathrm{v}_{14}$$\mathrm{v}_{15}$$\mathrm{v}_{16}$

12

12

14

15

17

$=\mathrm{S}\mathrm{p}\mathrm{a}\mathrm{n}\{1,y^{5}x\}$: $x^{4}+y^{5}+ax^{2}y^{3}$$f=x^{4}+y^{5}+ax^{2}y^{3}$ (of type (5, 4) of degree 20)
Partial derivatives : $f_{x}=4x^{3}+2axy^{3}$ , $f_{y}=5y^{4}+3ax^{2}y^{2}$
The standard base of: $\{y^{6},y^{4}x, 5y^{4}+3ay^{2}x^{2},2x^{3}+ay^{3}x\}$
Basis of the local ringand its degrees :$I=\langle f_{x}, f_{y}\rangle_{\mathit{0}}$$Ox,\mathit{0}/I$$m_{2}$$m_{3}$$m_{4}$$m_{5}$$m_{6}$$m_{7}$$m_{8}$$m_{9}$$m_{10}$$m_{11}$$m_{12}$$y$$x$$y^{2}$$yx$$x^{2}$$y^{3}$$y^{2}x$$yx^{2}$$y^{3}x$$y^{2}x^{2}$$y^{3}x^{2}$0458910

12

13

14

17

18

22

$m_{1}$

1
Basis of$\Sigma$:$\sigma_{1}=[\frac{1}{y^{4}x^{3}}+a(-\frac{3}{5}\frac{1}{y^{6}x}-\frac{1}{2}\frac{1}{yx^{5}})]$$\sigma_{4}=[\frac{1}{y^{2}x^{3}}]$,$\sigma_{5}=[\frac{1}{y^{3}x^{2}}]$,,$\sigma_{2}=[\frac{1}{y^{3}x^{3}}-\frac{3}{5}a\frac{1}{y^{5}x}]$$\sigma_{6}=[\frac{1}{y^{4}x}]$$\sigma_{10}=[\frac{1}{yx^{2}}]$,,$\sigma_{7}=[\frac{1}{yx^{3}}]$$\sigma_{11}=[\frac{1}{y^{2}x}]$

,

,

,

,

$\sigma_{3}=[\frac{1}{y^{4}x^{2}}-\frac{1}{2}a\frac{1}{yx^{4}}]$$\sigma_{8}=[\frac{1}{y^{2}x^{2}}]$,$\sigma_{9}=[\frac{1}{y^{3}x}]$,$\sigma_{12}=[\frac{1}{yx}]$

Degrees :$\sigma_{1}$-31Basis of$V$$\sigma_{2}$$\sigma_{3}$$\sigma_{4}$$\sigma_{5}$$\sigma_{6}$$\sigma_{7}$-27 -26 -23 -22 -20$\sigma_{8}$$\sigma_{10}$$U\mathfrak{g}$$\sigma_{11}$-19 -18 -17 -14 -13$\sigma_{12}$-9:$\mathrm{v}_{1}=10yx\partial_{x}-3ax^{2}\partial_{y}$$\mathrm{v}_{5}=y^{3}\partial_{y}$,,$\mathrm{v}_{2}=10y^{2}\partial_{y}+3ax^{2}\partial_{y}$$\mathrm{v}_{6}=y^{2}x\partial_{x}$$\mathrm{v}_{11}=y^{3}x\partial_{y}$,,$\mathrm{v}_{7}=y^{2}x\partial_{y}$$\mathrm{v}_{12}=y^{2}x^{2}\partial_{x}$

, V13

,

,

$\mathrm{v}_{3}=3x^{2}\partial_{x}+ay^{3}\partial_{\mathrm{g}}$$\mathrm{v}_{8}=yx^{2}\partial_{x}$$=y^{2}x^{2}\partial_{y}$

,

,

,

$\mathrm{v}_{9}=yx^{2}\partial_{y}$$\mathrm{v}_{14}=y^{3}x^{2}\partial_{x}$,$\mathrm{v}_{4}=2yx\partial_{y}-ay^{3}\partial_{x}$,$\mathrm{v}_{10}=y^{3}x\partial_{x}$$\mathrm{v}_{15}=y^{3}x^{2}\partial_{y}$Degrees :$\mathrm{v}_{1}$v2$\mathrm{v}_{3}$v4$\mathrm{v}_{5}$$\mathrm{v}_{6}$$\mathrm{v}_{7}$$\mathrm{v}_{8}$vg4455889910
Solution space $H$ :$H=\mathrm{S}\mathrm{p}\mathrm{a}\mathrm{n}\{1,y^{3}x^{2}\}$$\mathrm{v}_{10}$$\mathrm{v}_{11}$$\mathrm{v}_{12}$

12

13

13

V13

14

$\mathrm{v}_{14}$

17

V15

18

,

62
4.4.8

$W_{13}$: $x^{4}+xy^{4}+ay^{6}$(of tyPe (4, 3) of degree 16)
Partial derivatives : $f_{ax}=4x^{3}+y^{4}$ ,The standard base of:
Basis of the local ring $O_{X,O}/I$ and its degrees :$f=x^{4}+xy^{4}+ay^{6}$$f_{y}=4\mathrm{x}\mathrm{y}3+6ay^{5}$$I=\langle f_{x},f_{y})_{\mathit{0}}$$m_{1}$

1
0
Basis of$\Sigma$$\{y^{7},2\mathrm{y}3\mathrm{x}+3\mathrm{a}\mathrm{y}5,4\mathrm{x}3+y^{4}\}$$m_{2}$$m_{3}$$m_{4}$$m_{5}$$m_{6}$$m_{7}$$m_{8}$$m_{9}$$m_{10}$$m_{11}$$m_{12}$$m_{13}$$y$$x$$y^{2}$$yx$$x^{2}$$y^{3}$$y^{2}x$$yx^{2}$$y^{4}$$y^{2}x^{2}$$y^{5}$$y^{6}$

3

4

6

7

8

9

10

11

12

14

15

18

:

$\sigma_{1}=[\frac{1}{y^{7}x}-\frac{1}{4}\frac{1}{y^{3}x^{4}}+a(-\frac{3}{2}\frac{1}{y^{5}x^{2}}+\frac{3}{8}\frac{1}{yx^{5}})]$$\sigma_{4}=[\frac{1}{y^{5}x}-\frac{1}{4}\frac{1}{yx^{4}}]$$\sigma_{9}=[\frac{1}{y^{2}x^{2}}]$,,$\sigma_{5}=[\frac{1}{y^{2}x^{3}}]$$\sigma_{10}=[\frac{1}{y^{3}x}]$

,

,

,

,

$\sigma_{2}=[\frac{1}{y^{6}x}-\frac{1}{4}\frac{1}{y^{2}x^{4}}-\frac{3}{2}a\frac{1}{y^{4}x^{2}}]$,$\sigma_{6}=[\frac{1}{y^{3}x^{2}}]$$\sigma_{11}=[\frac{1}{yx^{2}}]$,,$\sigma_{7}=[\frac{1}{y^{4}x}]$$\sigma_{12}=[\frac{1}{y^{2}x}]$,$\sigma_{3}=[\frac{1}{y^{3}x^{3}}]$$\sigma_{8}=[\frac{1}{yx^{3}}]$,$\sigma_{13}=[\frac{1}{yx}]$

Degrees :$\sigma_{1}$$\sigma_{2}$$\sigma_{3}$$\sigma_{4}$$\sigma_{5}$$\sigma_{6}$$\sigma_{7}$$\sigma_{8}$$\sigma_{10}$$\sigma_{9}$$\sigma_{11}$-25 -22 -21 -19 -18 -17 -16 -15 -14 -13 -11
Basis of$V$$\sigma_{12}$$\sigma_{13}$

-10

-7

:
$\mathrm{v}_{1}=4yx\partial_{ae}+3y^{2}\partial_{y}-3ay^{3}\partial_{ae}$$\mathrm{v}_{4}=3y^{3}\partial_{l}+4x^{2}\partial_{y}$$\mathrm{v}_{8}=yx^{2}\partial_{y}$,,$\mathrm{v}_{2}=8x^{2}\partial_{x}-9ay^{3}\partial_{y}$,$\mathrm{v}_{5}=3y^{3}\partial_{y}+4y^{2}x\partial_{\mathrm{r}}$$\mathrm{v}_{9}=y^{4}\partial_{\mathrm{g}}$$\mathrm{v}_{13}=y^{5}\partial_{\mathrm{g}}$

,

,

,

$\mathrm{v}_{10}=y^{4}\partial_{y}$$\mathrm{v}_{14}=y^{5}\partial_{y}$

,

,

,

$\mathrm{v}_{3}=2yx\partial_{y}+3ay^{3}\partial_{y}$$\mathrm{v}_{6}=y^{2}x\partial_{y}$$\mathrm{v}_{11}=y^{2}x^{2}\partial_{x}$$\mathrm{v}_{15}=y^{6}\partial_{x}$

,

,

, V7

$=yx^{2}\partial_{ae}$$\mathrm{v}_{12}=y^{2}x^{2}\partial_{y}$

,

,

,

$\mathrm{v}_{16}=y^{6}\partial_{y}$Degrees :$\mathrm{v}_{1}$$\mathrm{v}_{2}$$\mathrm{v}_{3}$$\mathrm{v}_{4}$$\mathrm{v}_{5}$$\mathrm{v}_{6}$$\mathrm{v}_{7}$$\mathrm{v}_{8}$

3

4

4

5

Vg

6

7

7

8

8

Solution space H : H
4.4.9$Q_{10}$$\mathrm{v}_{10}$$\mathrm{v}_{11}$$\mathrm{v}_{12}$$\mathrm{v}_{13}$$\mathrm{v}_{14}$$\mathrm{v}_{15}$$\mathrm{v}_{16}$

9

10

11

11

12

14

15

$=\mathrm{S}\mathrm{p}\mathrm{a}\mathrm{n}\{1,y^{6}\}$: $x^{3}+y^{4}+yz^{2}+axy^{3}$$f=x^{3}+y^{4}+yz^{2}+axy^{3}$ (of tyPe (8, 6, 9) of degree 24)

Partial derivatives : $f_{l}=3x^{2}+ay^{3}$ , $f_{y}=4y^{3}+z^{2}+3axy^{2}$ , $f_{z}=2yz$
base of $I=(f_{l},$: $\{x^{4},4yx^{2}+3ax^{3},3x^{2}+ay^{3},zx^{2},zy, 12x^{2}-az^{2}-3a^{2}y^{2}x\}$
Basis of the local ring $O_{X,O}/I$ and its degrees:$\mathrm{q}\mathrm{h}\Psi \mathrm{S}\mathrm{t}\mathrm{a}\mathrm{n}\mathrm{d}\mathrm{a}\mathrm{r}\mathrm{d}$$h\rangle_{\mathit{0}}$$m_{1}$$m_{2}$$m_{3}$$m_{4}$$m_{5}$$m_{6}$$m_{7}$$m_{8}$$m_{9}$$m_{10}$

$y$

$x$

$z$

$y^{2}$$zx$$y^{3}$$y^{2}x$$y^{3}x$

0689

$yx$

12

14

17

18

20

26

1

Basis of I:$\sigma_{1}=[\frac{1}{zy^{4}x^{2}}-4\frac{1}{z^{3}yx^{2}}+a(-\frac{3}{4}\frac{1}{zy^{5}x}-\frac{1}{3}\frac{1}{zyx^{4}})+\frac{1}{4}a^{2}\frac{1}{zy^{2}x^{3}}]$$\sigma_{3}=[\frac{1}{zy^{4}x}-4\frac{1}{z^{3}yx}+\frac{1}{3}a\frac{1}{zyx^{3}}]$,$\sigma_{4}=[\frac{1}{z^{2}yx^{2}}]$,,$\sigma_{2}=[$ $\frac{1}{zy^{3}x^{2}}-\frac{3}{4}a\frac{1}{zy^{4}x}-\frac{1}{4}a^{2}\frac{1}{zyx’\backslash }$$\sigma_{5}=[\frac{1}{zy^{2}x^{2}}]$,$\sigma_{6}=[\frac{1}{zy^{3}x}]$

,

,

63
,

$\sigma_{7}=[\frac{1}{z^{2}yx}]$$\sigma_{8}=[\frac{1}{zyx^{2}}]$,,$\sigma_{9}=[\frac{1}{zy^{2}x}]$$\sigma_{10}=[\frac{1}{zyx}]$

Degrees :$\sigma_{10}$$\sigma_{9}$$\sigma_{8}$$\sigma_{7}$$\sigma_{6}$$\sigma_{5}$$\sigma_{4}$$\sigma_{3}$$\sigma_{2}$$\sigma_{1}$-49 -43 -41 -40 -37 -35 -32 -31 -29 -23
Basis of V :,$\mathrm{v}_{3}=4y^{2}\partial_{y}+a(3yx\partial_{y}+3zx\partial_{z}),\mathrm{v}_{4}=4yx\partial_{y}+6zx\partial_{z}-2ay^{3}\partial_{l}\mathrm{v}_{1}=z\partial_{y}+4y^{2}\partial_{z}+3ayx\partial_{z},\mathrm{v}_{2}=8yx\partial_{x}-a(4yx\partial_{y}+3zx\partial_{z})\mathrm{v}_{5}=zx\partial_{x},\mathrm{v}_{6}=4y^{3}\partial_{z}+3ay^{2}x\partial_{z},\mathrm{v}_{7}=zx\partial_{y}+4y^{2}x\partial_{z},’$$\mathrm{v}_{8}=y^{3}\partial_{y}$, Vg$=y^{2}x\partial_{l}$,$\mathrm{v}_{10}=y^{2}x\partial_{y}$,$\mathrm{v}_{11}=y^{3}x\partial_{z}$,$\mathrm{v}_{12}=y^{3}x\partial_{x}$,$\mathrm{v}_{13}=y^{3}x\partial_{y}$Degrees :v4$\mathrm{v}_{3}$$\mathrm{v}_{2}$$\mathrm{v}_{1}$V7$\mathrm{v}_{6}$$\mathrm{v}_{5}$$\mathrm{V}\mathfrak{g}$$\mathrm{v}_{8}$$\mathrm{v}_{10}$$\mathrm{v}_{11}$$\mathrm{v}_{12}$$\mathrm{v}_{13}$

14

17

18

20

36689911 12 12
Solution space $H$ :4.4.10$Q_{11}$$H=\mathrm{S}\mathrm{p}\mathrm{a}\mathrm{n}\{1,y^{3}x\}$: $x^{3}+y^{2}z+xz^{3}+az^{5}$$f=x^{3}+y^{2}z+xz^{3}+az^{5}$ (of type (6, 7, 4) of degree 18)

Partial derivatives : $f_{x}=3x^{2}+z^{3}$ , $f_{y}=2yz$ , $f_{z}=y^{2}+3xz^{2}+5az^{4}$: $\{x^{4}, yx^{2},9x^{3}-5ay^{2}x,y^{3},zx^{3}, zy, 3z^{2}x+y^{2}-15azx^{2},3x^{2}+z^{3}\}$The standard base of$O_{X,O}/I$
its
degrees :andBasis of the local ring
$I=\langle f_{ox},f_{y}\rangle \mathit{0}$$m_{6}zx$$m_{7}z^{3}$$m_{8}yx$04678101213$m_{1}1$

Basis of$\Sigma$$m_{2}z$$m_{3}x$$m_{5}z^{2}$$m_{4}y$$\sigma_{3}=[\frac{1}{zy^{3}x}-\frac{1}{3}\frac{1}{z^{3}yx^{2}}]$$\sigma_{7}=[\frac{1}{z^{3}yx}]$

,

,

,

$\sigma_{4}=[\frac{1}{zy^{2}x^{2}}]$$\sigma_{8}=[\frac{1}{zy^{2}x}]$,,$\sigma_{2}=[\frac{1}{z^{5}yx}-\frac{1}{3}\frac{1}{z^{2}yx^{3}}-\frac{5}{3}a\frac{1}{z^{3}yx^{2}}]$,$\sigma_{5}[\frac{1}{z^{4}yx}-\frac{1}{3}\frac{1}{zyx^{3}}]$$\sigma_{9}=[\frac{1}{zyx^{2}}]$,$\sigma_{10}=[\frac{1}{z^{2}yx}]$,$\sigma_{6}=[\frac{1}{z^{2}yx^{2}}]$,$\sigma_{11}=[\frac{1}{zyx}]$

egrees :$\sigma_{3}$$\sigma_{2}$$\sigma_{1}$-37 -33 -31
Basis of$V$$\sigma_{4}$$\sigma_{6}$$\sigma_{5}$:$\mathrm{v}_{3}=2yx\partial_{y}+2zx\partial_{z}-5az^{2}x\partial_{x}$$\mathrm{v}_{5}=yx\partial_{x}$,,$\mathrm{v}_{10}=z^{4}\partial_{x}$,,$\mathrm{v}_{2}=6zx\partial_{x}+4z^{2}\partial_{z}-5ayx\partial_{y}$,$\mathrm{v}_{7}=3z^{2}x\partial_{x}+2z^{3}\partial_{z}$$\mathrm{v}_{11}=z^{4}\partial_{z}$,$\mathrm{v}_{12}=z^{5}\partial_{y}$egrees :$\mathrm{v}_{1}$v2$\mathrm{v}_{3}$v4$\mathrm{v}_{5}$$\mathrm{v}_{6}$v73466778
Solution space $H$ :$H=\mathrm{S}\mathrm{p}\mathrm{a}\mathrm{n}\{1, z^{5}\}$$\sigma_{11}$$\sigma_{10}$,$\mathrm{v}_{4}=6zx\partial_{z}+2z^{3}\partial_{x}-a(15z^{2}x\partial_{x}+10z^{2}x\partial_{x})$$\mathrm{v}_{6}=3z^{2}x\partial_{y}+5az^{4}\partial_{y}$Vg $=z^{2}x\partial_{z}$ ,$\sigma_{9}$$\sigma_{8}$$\sigma_{7}$-30 -29 -27 -25 -24 -23 -21 -17$\mathrm{v}_{1}=3zx\partial_{y}+y\partial_{z}+5az^{3}\partial_{y}$$\mathrm{D}$

20

416

:

$\sigma_{1}=[\frac{1}{z^{6}yx}+\frac{1}{zy^{3}x^{2}}-\frac{1}{3}\frac{1}{z^{3}yx^{3}}+a(-\frac{5}{3}\frac{1}{z^{4}yx^{2}}+\frac{5}{9}\frac{1}{zyx^{4}})]$$\mathrm{D}$$m_{11}z^{5}$$m_{10}z^{4}$$z^{2}xm_{9}$$\mathrm{v}_{8}$

$9$

vg

10

,

,

$\mathrm{v}_{8}=z^{4}\partial_{y}-yx\partial_{z}$$\mathrm{v}_{13}=z^{5}\partial_{x}$,$\mathrm{v}_{14}=z^{5}\partial_{z}$$\mathrm{v}_{10}$$\mathrm{v}_{11}$$\mathrm{v}_{12}$$\mathrm{v}_{13}$$\mathrm{v}_{14}$

10

12

13

14

16

,

,

,

64
4.4.11$Q_{12}$: $x^{3}+y^{5}+yz^{2}+axy^{4}$$f=x^{3}+y^{5}+yz^{2}+axy^{4}$ (of type (5, 3, 6) of degree 15)
Partial derivatives : $f_{l}=3x^{2}+ay^{4}$ , $f_{y}=5y^{4}+z^{2}+4axy^{3}$ , $f_{z}=2yz$
The standard base of: $\{x^{4},5yx^{2}+4ax^{3},3x^{2}+\mathrm{a}\mathrm{y}4, zx^{2}, zy, 15x^{2}-az^{2}-4a^{2}y^{3}x\}$
Basis of the local ring $O_{X,O}/I$ and its degrees :$I=(f_{l}, \mathrm{f}\mathrm{y})0$$m_{1}$

1$m_{2}$$m_{3}$$m_{4}$$m_{5}$$m_{6}$$m_{7}$$m_{8}$$m_{9}$$m_{10}$$m_{11}$$m_{12}$

$y$

$x$

$z$

$y^{2}$$yx$$y^{3}$$zx$$y^{2}x$$y^{4}$$y^{3}x$$y^{4}x$

11

12

14

17

035668911
Basis of:$\Sigma$$\sigma_{1}=[\frac{1}{zy^{5}x^{2}}-5\frac{1}{z^{3}yx^{2}}+a(-\frac{4}{5}\frac{1}{zy^{6}x}-\frac{1}{3}\frac{1}{zyx^{4}})+\frac{4}{15}a^{2}\frac{1}{zy^{2}x^{3}}]$$\sigma_{3}=[\frac{1}{zy^{5}x}-5\frac{1}{z^{3}yx}-\frac{a}{3}\frac{1}{zyx^{3}}]$$\sigma_{8}=[\frac{1}{zy^{3}x}]$,,$\sigma_{4}=[\frac{1}{zy^{3}x^{2}}]$,$\sigma_{9}=[\frac{1}{z^{2}yx}]$,,$\sigma_{2}=[\frac{1}{zy^{4}x^{2}}-\frac{4}{5}a\frac{1}{zy^{5}x}+\frac{4}{15}a^{2}\frac{1}{zyx^{3}}]$,$\sigma_{5}=[\frac{1}{z^{2}yx^{2}}]$$\sigma_{10}=[\frac{1}{zyx^{2}}]$,$\sigma_{6}=[\frac{1}{zy^{4}x}]\sigma_{7}=[\frac{1}{zy^{2}x^{2}}]$$\sigma 11=[\frac{1}{zy^{2}x}]$

,

,

,

$\sigma_{12}=[\frac{1}{zyx}]$

Degrees :$\sigma_{1}$$\sigma_{2}$$\sigma_{3}$$\sigma_{4}$$\sigma_{5}$$\sigma_{6}$$\sigma_{7}$$\sigma_{8}$$\sigma_{10}$$\sigma_{9}$$\sigma_{11}$$\sigma_{12}$-31 -28 -26 -25 -25 -23 -22 -20 -20 -19 -17 -14
Basis of:$V$$\mathrm{v}_{1}=5yx\partial_{x}-a(2yx\partial_{y}+2zx\partial_{z})$V3 $=5y^{2}\partial_{y}+a(4yx\partial_{y}+6zx\partial_{z})$ ,$\mathrm{v}_{5}=y^{3}\partial_{y}$,$\mathrm{v}_{9}=y^{2}x\partial_{y}$$\mathrm{v}_{6}=y^{2}x\partial_{l}$,$\mathrm{v}_{2}$$\mathrm{v}_{3}$$\mathrm{v}_{4}$$\mathrm{v}_{2}=z\partial_{y}+5y^{3}\partial_{z}+4ay^{2}x\partial_{z}$

V5

,

V7

$\mathrm{v}_{6}$,$\mathrm{v}_{7}=zx\partial_{a}$$\mathrm{v}_{14}=y^{4}x\partial_{z}$$\mathrm{v}_{8}$,$\mathrm{V}\mathfrak{g}$,$\mathrm{v}_{8}=5y^{4}\partial_{z}+4ay^{3}x\partial_{z}$$\mathrm{v}_{11}=y^{4}\partial_{y}$$\mathrm{v}_{15}=y^{4}x\partial_{l}$$\mathrm{v}_{10}$$\mathrm{v}_{11}$,,4.4.12$S_{11}$,,$\mathrm{v}\mathrm{i}_{2}\overline{\wedge}y^{3}x\partial_{g}$,$\mathrm{v}_{16}=y^{4}x\partial_{y}$$\mathrm{v}_{12}$$\mathrm{v}_{13}$33356666889911
Solution space H : H,$\mathrm{v}_{4}=3yx\partial_{y}+6zx\partial_{z}-2ay^{4}\partial_{\mathrm{g}}$$\mathrm{v}_{10}=zx\partial_{y}+5y^{3}x\partial_{z}$$\mathrm{v}_{13}=y^{3}x\partial_{y}$$\mathrm{v}_{1}$,,$\mathrm{v}_{14}$$\mathrm{v}_{15}$$\mathrm{v}_{16}$

11

12

14

$=\mathrm{S}\mathrm{p}\mathrm{a}\mathrm{n}\{1,y^{4}x\}$: $x^{4}+y^{2}z+xz^{2}+ay^{2}x^{2}$(of type (4, 5, 6) of degree 16)
Partial derivatives : $f_{l}=4x^{3}+z^{2}+2ay^{2}x$ ,The standard base of $I=$:
$f=x^{4}+y^{2}z+xz^{2}+ay^{2}x^{2}$$f_{y}=2zy+2\mathrm{a}\mathrm{y}\mathrm{x}2$, $f_{z}=y^{2}+2zx$$(f_{x}, \mathrm{f}\mathrm{y})0$$\{4x^{5}+5a^{2}x^{6},4yx^{3}+5a^{2}yx^{4},8x^{4}+5ay^{2}x^{2},y^{3}-2ayx^{3},2zx+y^{2}, zy+ayx^{2},4x^{3}+z^{2}+2ay^{2}x\}$

Basis of the local ring

$O_{X,\mathit{0}}/I$$m_{3}$$m_{4}$

$x$

$y$

$z$

$m_{5}$$m_{7}$$m_{8}$$m_{9}$$m_{10}$$m_{11}$$yx$$y^{2}$$x^{3}$$yx^{2}$$y^{2}x$$y^{2}x^{2}$04568910

12

13

14

18

$\backslash 1$$\Sigma$$m_{2}$$m_{6}$$m_{\mathrm{t}}1$

Basis ofand its degrees :$x^{2}$:$\sigma_{1},\cdot=[\frac{1}{zy^{3}x^{3}}+2\frac{1}{z^{4}yx}-\frac{1}{2}\frac{1}{z^{2}yx^{4}}+a(-\frac{1}{z^{2}y^{3}x}+\frac{1}{2}\frac{1}{z^{3}yx^{2}}-\frac{5}{8}\frac{1}{zyx^{5}})]$$\sigma_{2}=[\frac{1}{zy^{3}x^{2}}-\frac{1}{2}\frac{1}{z^{2}yx^{3}}-\frac{1}{2}a\frac{1}{zyx^{4}}]$$f \sigma_{5}=[-\frac{1}{zy^{3}x}+\frac{1}{2}\frac{1}{z^{2}yx^{2}}]$,,$\sigma_{3}=[\frac{1}{zy^{2}x^{3}}-a\frac{1}{z^{2}y^{2}x}]$$\sigma_{6}=[\frac{1}{zy^{2}x^{2}}]$

,

,

,

$\sigma_{4}=[\frac{1}{z^{3}yx}-\frac{1}{4}\frac{1}{zyx^{4}}]$$\sigma_{7}=[\frac{1}{zyx^{3}}]$,$\sigma_{8}=[\frac{1}{z^{2}yx}]$

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,

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65
$\sigma_{9}=[\frac{1}{zy^{2}x}]$

Degrees :,$\sigma_{10}=[\frac{1}{zyx^{2}}]$$\sigma_{11}=[\frac{1}{zyx}]$.
-33 -29 -28 -27 -25 -24 -23 -21 -20 -19 -15$\sigma_{2}$$\sigma_{1}$Basis of,$V$$\sigma_{3}$$\sigma_{5}$$\sigma_{4}$$\sigma_{6}$$\sigma_{7}$$\sigma_{8}$$\sigma_{10}$$\sigma_{9}$$\sigma_{11}$:$\mathrm{v}_{1}=64x^{2}\partial_{l}+32y^{2}\partial_{z}+144ax^{3}\partial_{z}-65a^{2}y^{2}x\partial_{z}$$\mathrm{v}_{2}=16yx\partial_{y}-16y^{2}\partial_{z}-48ax^{3}\partial_{z}+15a^{2}y^{2}x\partial_{z}$$\mathrm{v}_{4}=yx\partial_{x}$,$\mathrm{v}_{5}=4y^{2}\partial_{x}+48x^{3}\partial_{z}-21ay^{2}x\partial_{z}$,$\mathrm{v}_{7}=2x^{3}\partial_{x}+y^{2}x\partial_{z}$,$\mathrm{v}_{11}=y^{2}x\partial_{x}$$\mathrm{v}_{8}=yx^{2}\partial_{y}-y^{2}x\partial_{z}$$\mathrm{v}_{12}=y^{2}x^{2}\partial_{z}$

,

,

,

,

,

$\mathrm{v}_{3}=y^{2}\partial_{y}$,$\mathrm{v}_{6}=2x^{3}\partial_{y}+yx^{2}\partial_{z}$$\mathrm{v}_{9}=yx^{2}\partial_{x}$$\mathrm{v}_{13}=y^{2}x^{2}\partial_{y}$

,

,

,

$\mathrm{v}_{10}=y^{2}x\partial_{y}$,$\mathrm{v}_{14}=y^{2}x^{2}\partial_{l}$Degrees :$\mathrm{v}_{2}$$\mathrm{v}_{1}$$\mathrm{v}_{3}$$\mathrm{v}_{4}$$\mathrm{v}_{5}$$\mathrm{v}_{6}$$\mathrm{v}_{7}$$\mathrm{v}_{11}$$\mathrm{v}_{10}$$\mathrm{v}_{9}$$\mathrm{v}_{8}$$\mathrm{v}_{12}$$\mathrm{v}_{13}$$\mathrm{v}_{14}$

12

13

14

445567889910
Solution space $H$ :4.4.13$S_{12}$$H=\mathrm{S}\mathrm{p}\mathrm{a}\mathrm{n}\{1,y^{2}x^{2}\}$: $x^{2}y+y^{2}z+xz^{3}+az^{5}$(of type (4, 5, 3) of degree 13)$f_{z}=y^{2}+3xz^{2}+5az^{4}$
Partial derivatives : $f_{x}=2xy+z^{3}$ ,The standard base of:$f=x^{2}y+y^{2}z+xz^{3}+az^{5}$$f_{y}=x^{2}+\mathrm{Z}\mathrm{x}\mathrm{z}2$$I=\langle f_{x}, f_{y}\rangle_{\mathit{0}}$$\{x^{4}, yx^{3}, y^{2}x,39yx^{2}-20ay^{3},13yx^{2}-10azx^{3}, x^{2}+2zy, 3z^{2}x+y^{2}+5ax^{3},2yx+z^{3}\}$Basis of the local ring $O_{X,O}/I$ and its degrees :$m_{1}1$$m_{2}z$$m_{3}x$$m_{5}z^{2}$$m_{4}y$$m_{6}zx$$m_{7}x^{2}$$z^{2}xm_{9}$$m_{8}z^{3}$$m_{10}zx^{2}$$m_{11}z^{4}$$m_{12}z^{5}$

11

12

15

0345678910
Basis of$\Sigma$:$\sigma_{1}=[\frac{1}{z^{6}yx}+\frac{3}{2}\frac{1}{zy^{4}x}-\frac{1}{2}\frac{1}{z^{3}y^{2}x^{2}}+\frac{1}{z^{2}yx^{4}}+a(-\frac{5}{13}\frac{1}{z^{2}y^{3}x}-\frac{20}{13}\frac{1}{z^{4}yx^{2}}+\frac{10}{13}\frac{1}{zy^{2}x^{3}})]$$\sigma_{2}=[\frac{1}{z^{5}yx}-\frac{1}{2}\frac{1}{z^{2}y^{2}x^{2}}+\frac{1}{zyx^{4}}-\frac{5}{3}a\frac{1}{z^{3}yx^{2}}]$,$\sigma_{3}=[\frac{1}{z^{2}yx^{3}}-\frac{1}{2}\frac{1}{z^{3}y^{2}x}]$$\sigma_{5}=[\frac{1}{z^{4}yx}-\frac{1}{2}\frac{1}{zy^{2}x^{2}}]$,$\sigma_{6}=[\frac{1}{zyx^{3}}-\frac{1}{2}\frac{1}{z^{2}y^{2}x}]$$\sigma_{9}=[\frac{1}{zy^{2}x}]$,$\sigma_{10}=[\frac{1}{zyx^{2}}]$

,

,

,

$\sigma_{4}=[\frac{1}{z^{3}yx^{2}}-3\frac{1}{zy^{3}x}]$,$\sigma_{7}=[\frac{1}{z^{2}yx^{2}}]$$\sigma_{11}=[\frac{1}{z^{2}yx}]$,,$\sigma_{8}=[\frac{1}{z^{3}yx}]$,$\sigma_{12}=[\frac{1}{zyx}]$

Degrees :$\sigma_{1}$$\sigma_{2}$$\sigma_{4}$$\sigma_{3}$$f\sigma_{f}$$\sigma_{5}$$\sigma_{7}$-27 -24 -23 -21 -21 -20
Basis of$V$$-\mathrm{i}9$$\sigma_{10}$$\sigma_{9}$$\sigma_{8}$$\sigma_{12}$$\sigma_{11}$-18 -17 -16 -15 -12:,$\mathrm{v}_{1}=39z^{2}\partial_{x}+39zx\partial_{y}+26y\partial_{z}+130az^{3}\partial_{y}+200a^{2}zx^{2}\partial_{y}$$\mathrm{v}_{2}=104zx\partial_{x}-65x^{2}\delta_{y}+78z^{2}\partial_{z}+260az^{2}x\partial_{y}+400a^{2}z^{4}\partial_{y}$$\mathrm{v}_{3}=13z^{3}\partial_{y}+26zx\partial_{z}+80azx^{2}\partial_{y}$,$\mathrm{v}_{5}=78z^{2}x\partial_{y}+13x^{2}\partial_{z}+110az^{4}\partial_{y}$$\mathrm{v}_{7}=3zx^{2}\partial_{y}-2z^{3}\partial_{z}$$\mathrm{v}_{11}=zx^{2}\partial_{z}$,Degrees :$\mathrm{v}_{1}$

v2

,

,

$\mathrm{v}_{13}=z^{4}\partial_{z}$.
$\mathrm{v}_{3}$$\mathrm{v}_{4}$,,,$\mathrm{v}_{8}=2z^{2}x\partial_{x}+zx^{2}\partial_{y}$$\mathrm{v}_{12}=z^{4}\partial_{x}$$\mathrm{v}_{5}$$\mathrm{v}_{6}$$\mathrm{v}_{7}$$\mathrm{v}_{8}$vg$\mathrm{v}_{6}=26z^{3}\partial_{x}-13x^{2}\partial_{z}+50az^{4}\partial_{y}$$\mathrm{v}\mathfrak{y}=z^{4}\partial_{y}+2z^{2}x\partial_{z}$$\mathrm{v}_{14}=z^{5}\partial_{y}$,,$H=\mathrm{S}\mathrm{p}\mathrm{a}\mathrm{n}\{1,z^{5}\}$,,$\mathrm{v}_{10}=zx^{2}\partial_{x}-z^{4}\partial_{y}$$\mathrm{v}_{15}=z^{5}\partial_{x}$,$\mathrm{v}_{12}$$\mathrm{v}_{14}$$\mathrm{v}_{16}=z^{5}\partial_{z}$‘$\mathrm{v}_{10}$$\mathrm{v}_{11}$$\mathrm{v}_{13}$234455667788810
Solution space $H$ :,$\mathrm{v}_{4}=13x^{2}\partial_{x}-13z^{3}\partial_{y}-30azx^{2}\partial_{y}$$\mathrm{v}_{15}$$\mathrm{v}_{16}$

11

12

,

66
4.4.14: $x^{3}+y^{3}+z^{4}+axyz^{2}$$U_{12}$$f=x^{3}+y^{3}+z^{4}+axyz^{2}$ (of tyPe (4, 4, 3) of degree 12)

Partial derivatives : $f_{x}=3x^{2}+ayz^{2}$ , $f_{y}=3y^{2}+axz^{2}$ , $f_{z}=4z^{3}+2axyz$
The standard base of $I=(f_{x},$: $\{x^{4},yx^{2},y^{2}x,x^{3}-y^{3}, zx^{2},zy^{2},3y^{2}+az^{2}x,3x^{2}+az^{2}y, 2z^{3}+azyx\}$
Basis of the local ringand its degrees :$f_{y}\rangle$$0$$\mathrm{O}\mathrm{x},0/\mathrm{I}$$m_{1}$$m_{2}$$m_{3}$$m_{4}$

1
0344
$z$

Basis of$\Sigma$$m_{5}$$z^{2}$

$x$

$y$

6

$m_{6}$$m_{7}$$m_{8}$$m0$$m_{10}$$m_{11}$$m_{12}$

$zy$

$zx$

$yx$

$z^{2}y$$z^{2}x$$zyx$$z^{2}yx$

7

7

8

10

10

11

14

:
$\sigma_{1}=[\frac{1}{z^{3}y^{2}x^{2}}-\frac{1}{3}a\frac{1}{zyx^{4}}+a(-\frac{1}{2}\frac{1}{z^{5}yx}-\frac{1}{3}\frac{1}{zy^{4}x})]$$\sigma_{3}=[\frac{1}{z^{3}y^{2}x}-\frac{1}{3}a\frac{1}{zyx^{3}}]$$\sigma_{7}=[\frac{1}{z^{2}y^{2}x}]$,,$\sigma_{8}=[\frac{1}{z^{3}yx}]$$\sigma_{4}=[\frac{1}{z^{3}yx^{2}}-\frac{1}{3}a\frac{1}{zy^{3}x}]$,$\sigma_{9}=[\frac{1}{zyx^{2}}]$

,

,

,

,

$\sigma_{2}=[\frac{1}{z^{2}y^{2}x^{2}}-\frac{1}{2}a\frac{1}{z^{4}yx}]$$\sigma_{5}=[\frac{1}{zy^{2}x^{2}}]$$\sigma_{10}=[\frac{1}{zy^{2}x}]$,,$\sigma_{6}=[\frac{1}{z^{2}yx^{2}}]$$\sigma_{11}=[\frac{1}{z^{2}yx}]$,,$\sigma_{12}=[\frac{1}{zyx}]$

Degrees :$\sigma_{2}$$\sigma_{1}$$\sigma_{3}$$\sigma_{4}$$\sigma_{5}$$\sigma_{6}$$\sigma_{7}$$\sigma_{8}$$\sigma_{9}$$\sigma_{10}$$\sigma_{12}$$\sigma_{11}$-25 -22 -21 -21 -19 -18 -18 -17 -15 -15 -14 -11
base of V :$\mathrm{v}_{1}=6zx\partial_{l}-ayx\partial_{z}$,$\mathrm{v}_{2}=6zy\partial_{y}-ayx\partial_{z}$,$\mathrm{v}_{3}=-3z^{2}\partial_{z}-ayx\partial_{z}$,$\mathrm{v}_{4}=6yx\partial_{l}-az^{2}x\partial_{y}$$\mathrm{v}_{8}=z^{2}y\partial_{y},\mathrm{v}_{9}=z^{2}x\partial_{l},\mathrm{v}_{10}=z^{2}y\partial_{z},\mathrm{v}_{11}=z^{2}x\partial_{z},\mathrm{v}_{12}=zyx\partial_{y},\mathrm{v}_{13}=zy’ x\partial_{x}\mathrm{v}_{5}=-3zx\partial_{z}+az^{2}y\partial_{l},\mathrm{v}_{6}=3zy\partial_{z}-az^{2}x\partial_{y},\mathrm{v}_{7}=-6yx\partial_{y}+az^{2}y\partial_{l}\mathrm{v}_{14}=zyx\partial_{z},\mathrm{v}_{15}=z^{2}yx\partial_{y},\mathrm{v}_{16}=z^{2}yx\partial_{l},\mathrm{v}_{17}=z^{2}yx\partial_{z}$,,Degrees :$\mathrm{v}_{1}$v2$\mathrm{v}_{3}$v4$\mathrm{v}_{5}$$\mathrm{v}_{6}$$\mathrm{v}_{7}$$\mathrm{v}_{8}$$\mathrm{v}_{9}$$\mathrm{v}_{10}$$\mathrm{v}_{11}$$\mathrm{v}_{12}$$\mathrm{v}_{13}$$\mathrm{v}_{14}$V15$\mathrm{v}_{16}$3334444667777810
Solution space H : H

10

V17

11

$=\mathrm{S}\mathrm{p}\mathrm{a}\mathrm{n}\{1,z^{2}yx\}$

References
V.I. ARNOLD, S.M. GUSEIN-ZADE and A.N. VARCHENKO, Singularities of Differentiate Maps
Volume I, Monographs in Mathematics Vol. 82, Birkh\"auser, Boston, 1985.
[2] K. SAITO, $Quafi.hom\eta ene$ isolierte singularitdten von hyperfldchen, Invent. Math. 14 (1971), 123142.
[1][3] 中村弥生, 田島慎一, Unimodal 例外型特異点に B ける代数的局所コホモロジー類, 京都大学数理解析
研究所講究録, 1211 (2001), 155-165.
[4] K. SAITO, Einfach-eliptische$Sing|dar\dot{|}t\text{\"{a}} ten$

Invent. Math. 23 (1974), 289-325.

[5] 田烏慎一, 中村弥生, Milnor algebra に付随した holonomic 系について, 京都大学数理解析研究所講究録, 1212(2 叩 y, $13\succ 143$ .

[6]

S. TAJIMA, T. OAKU and Y. NAKAMURA, Multidimensional local residues and holonomic Dmodules, Surikaiseki Kenkyusho Koyiiroku, RIMS Kyoto Univ. 1033 (1998), 59-70