数理解析研究所講究録 1211 巻 2001 年 155-165

155
Unimodal 例外型特異点における代数的局所コホモロジー類
中村 弥生 (Yayoi Nakamura) お茶の氷女子大学大学院 *田島 慎一 (Shinichi Tajima) 新潟大学工学部情報工学科 \dag er

1

序

孤立特異点を持つ擬斉次多項式 $f=f(x_{1}, \ldots, x_{n})$ に対し, 原点に台を持っ代数的局所コホモロジー類を考える. 但し, $f_{ae_{j}}=$, $j=1$ ,, である. コホモロジー類の annihilator }こついて, 次が成り立つ.
のを quasiweight
変数擬斉次多項式とする.
,, quasidegree
Fact : $f=f(x_{1}$ ,
を考える.
,
,
に対し, 原点に台を持つ代数的局所コホモロジー類の微分カ I 群としての annihilating ideal を $Ann$ とおくと,$\partial f/$$[1/f_{x_{1}}\cdots f_{xx_{n}}]\in \mathcal{H}_{[0]}^{n}(Ox)$$\ldots$$n$$[1/f_{x_{1}}\cdots f_{x_{n}}]$$\ldots$$\partial f/\partial x_{1}$$\ldots$$(\alpha_{1}, \ldots, \alpha_{n})$$x\mapsto$$d_{w}$$n$$f_{x_{n}}=\partial f/\partial x_{n}$$f_{\mathrm{g}_{1}}=$$[1/f_{l_{1}}\cdots f_{x_{n}}]$$[1/f_{x_{1}}\cdots f_{x_{n}}]$$Ann=\langle f_{x_{1}}, \ldots, f_{x_{n}}, P\rangle$が成り立つ. (旦し,$P= \alpha_{1}x_{1}\frac{\partial}{\partial x_{1}}+\cdots+\alpha_{n}x_{n}\frac{\partial}{\theta x_{n}}+nd_{w}-(\alpha_{1}+\cdots+\alpha_{n})$である.

( $Ann$ のグレブナ基底については [2] を参照されたい. )この事実を用いると, 微分作用素,,, $P$ を用いて代数的局所コホモロジー類を特徴付けろことができる ([4], [5] 参明). つまり,が擬斉次多項式の場合, 代数的局所コホモロジー類を特徴付ける微分作用素は euler 型で与えることができる. それに比べ, が半擬斉次多項式の場合, 一般に, 1 階の微分作用素では代数的局所コホモロジー類を特徴付けることはできない.
我々は, V.I Arnol’d による分類 ([1]) に従 , Unimodal 例外型孤立特異点 $(E_{12},$,,,,, Qio,, $\# 12$ ,, $\# 12$ ,,,) に付随する代数的局所 ホモロジー類の annihilator を計算した. その結果, これらの場合, 代数的局所コホモロジー類を特徴付けるには 2 階の微分作用素が必要とな$f_{x_{1}}$$\ldots$$[1/f_{l_{1}}\cdots f_{l_{n}}]$$f_{x_{n}}$$f$$[1/f_{x_{1}}\cdots f_{x_{n}}]$$f$$\backslash$$\ovalbox{\tt\small REJECT}$$\mathrm{E}1\mathrm{A}$$Q_{11}$$W_{13}$$Z_{13}$$S_{11}$$\# 12$$U_{12}$$\mathrm{E}1\mathrm{A}$$Z_{11}$$Z_{12}$$\text{コ}$ることが明らかになった.

本稿では, まず初めに, \S 1 で型特異点に対してコホモロジー類の計算法を述べ, \S 2 で主結果を与え
で型の場合につ\S 3 Unimodal 特異点に関する計算結果を与え, 最後に \S 4 で Bimodal 特異点である$E_{12}$る.

$E_{18}$

いて述べる.

2

代数的局所コホモロジー類の計算例 ($E_{12}$型)

型 Unimodal singularity を考えで生成される上のイデアノレ, $f_{y}=\partial f/\partial y=5xy^{4}+7y^{6}$ とおく.
,
でのグレブナ基底は Gb $=\{5x^{3}+7y^{2}x^{2},125x^{4}-1029yx^{3},3x^{2}+$:こ対し, 全次数辞書式順字$-21yx^{2}+5y^{4}x\}$ で与えられる.
の準素イデア’ レ分解は $I=I_{1}\cap I_{2}$ , $I_{1}=\langle 25y+147,3125\mathrm{x} +151263\rangle$ ,$I_{2}=\langle y^{8},5y^{4}x+7y^{6},3x^{2}+y^{5}\rangle$ で与えられる.
$Y=$ $\{(x, y)\in X | f_{x}=f_{y}=0\}$ とおくと, $Y=$$\{3125x+151263=25y+147=0\}\cup\{x =y=0\}$ であり, 原点の重複度は 12 である.に台を持つ代数的局所コホモロジー類 $[1/f_{x}f_{y}]$ に対し, $[1/f_{x}f_{y}]$ を annihilate する高々 階の微分作
用素の生成十る左イデアルを $Ann^{(j)}$ とおく. $Ann^{(1)}$ を求めると,を得る. 但し,$P_{1}=(5xy+7y^{3})\partial_{y}+20x$ $+42y^{2}$ ,$P_{2}=(3500y^{2}+2058y)\mathrm{x}\partial_{x}+$ ( (-1000y -735)x+7203y2)$-4000\mathrm{a};+7700y^{2}+84378y$である. これらの作用素を用いて原点に台を持つ代数的局所コホモロジー類)
計参照). 方程式几 $\sigma=f_{y}\sigma=P_{1}\sigma=P_{2}\sigma=0$ ,,$X=\vee\neg 2$

る.

上の半擬斉次多項式 $f(x, y)=x^{3}+xy^{5}+y^{7}$ で与えられる$f_{x}=\partial f/\partial x=3x^{2}+y^{5}$$f_{x}$$I=\langle f_{x}, f_{y}\rangle$$x$$E_{12}$$f_{y}$$\mathcal{O}_{X}$$\succ y$$I$$y^{5},$$Y$$j$$Ann^{(1)}=\langle f_{l}, f_{y}, P_{1}, P_{2}\rangle$$(\partial_{x}=\partial/\partial x, \partial_{y}=\partial/\partial y)$\sigma 12=[1/[1x/yf]x を解 (\ near ow 0,0 ニ$\mathit{0}$$\not\equiv<\mathrm{E}\text{を}[succeq]_{1\sim\ddagger 9}-$*nakamura@math.ocha ac.jp
taj ima@geb ge niigata-u ac iP$\mathrm{t}${現 [4]$\sigma \text{の}\not\equiv\ovalbox{\tt\small REJECT} \mathrm{T}6$$|\partial(f_{x}, f_{y})/\partial(x, y)|\sigma=$

156$\sigma=[$$\underline{\cap?}\frac{1}{x\mu_{8}}-\frac{1220703125}{1483273860320763,3125},\frac{1}{x?^{2}}+\frac{48828125}{10090298369529,12515},\frac{1}{xy_{1}^{3}}-125\frac{1953125}{68641485507}\frac{1}{xy^{4}}$9765625$+_{\overline{466948881}} \frac{1}{xy^{5}1}-\overline{3176523}\overline{xy^{6}}\overline{21609}x+--\frac{1}{x^{2}y}\overline{6\mathrm{f}_{525111}^{7}}\overline{147}\overline{xy^{8}}\overline{1441471195647}$$+ \frac{390625}{9805926501}\overline{x^{2}y^{2}}-\frac{15625}{66706983}\frac{1}{x^{2}y^{3}}+\frac{1}{x^{2}y^{4}}-\overline{453789}\overline{3087}\overline{x^{2}y^{5}}+\overline{21}\overline{x^{2}y^{6}}$

3125

1125

15111

$+_{\overline{9529569}\overline{x^{3}y}}-\overline{64827}\overline{xy^{2}}+\overline{441}\overline{x^{3}y^{3}}\overline{63}\overline{x^{4}y}-]$$\text{を_{}(}’\doteqdot 6$.

$t_{\mathrm{J}^{1}}6$.

$’\supset\yen \mathrm{V}2$$f\sim.$$\not\equiv$

,

,

ffffl

$Ann^{(1)}\text{を}$$l$$\backslash farrow.\vec{\mathrm{q}}\Rightarrow+\Leftrightarrow\nabla\#\mathrm{f},\overline{\mathcal{T}}\grave{\grave{J}}\triangleright F\ovalbox{\tt\small REJECT} \text{数}\}_{arrow}^{-}7\mathrm{B}^{\backslash }\pm’T6\oplus’*[1/xy]\sigma)\Gamma+_{\backslash }\text{数}\emptyset\grave{\grave{\}}}\Re\ovalbox{\tt\small REJECT}\dot{\mathrm{b}}^{f_{jl}}$$\backslash \underline{\nu}r_{y\grave{\mathrm{i}}’}\sim A$$Ann^{(1)}\text{を}\overline{J\mathrm{F}_{\backslash }}\text{点_{}1\backslash }\mathfrak{l}^{-}.\text{局}\overline{\mathrm{p},}|\mathrm{i}l\mathrm{b}\mathrm{b}$$\gamma\sim.\mathrm{b}$$\text{の}[] \mathrm{f}$,$\langle y^{8},5y^{4}x+7y^{6},3x^{2}+y^{5}, P_{0},, {}_{1}P_{0,2}\rangle’arrow t\triangleright;6$$P_{0,1}=$$(5yx +7y^{3})\partial_{y}+20x$ $+42y^{2}$$P_{0,2}=$$470880590578020yx\partial_{l}+(-16817163949215x+164808206702307y^{2})\partial_{y}$-C$\text{あ}6$.

$\grave{;}\mathrm{A}\}^{-}.$

$P=$

.

$\{\underline{\mathrm{B}}\mathrm{b}$,$-9765625000x^{3}+(-80390625000y+236348437500)x^{2}$$+(56273437500y^{3}-330887812500y^{2}+1945620337500y-11440247584500)x$$+463242937500y^{4}-2723868472500y^{3}+16016346618300y^{2}+1930610421369882y$$P_{0,1}$,,’$P_{0,2}\text{の}\hslash^{\nearrow}\acute,\hslash\backslash \dot{\downarrow}\supset,- \mathrm{p}_{\backslash \text{ロ}}$$Ann^{(2)}\text{を}\mathfrak{R}d)$$6\underline{\mu}$,)$\nearrow\backslash \backslash \backslash \backslash /^{\backslash }i\#_{\backslash }D_{X}/Ann^{(1)}\mathit{0}$$Ann^{(2)}=\langle f_{l}, f_{y}, P\rangle\xi j\acute{(}\ovalbox{\tt\small REJECT} 6$.

$\ovalbox{\tt\small REJECT} \mathrm{A}_{1\backslash }\}^{-}.k^{\backslash }t\mathrm{e}6\ovalbox{\tt\small REJECT}\dagger\Xi \mathrm{E}7_{J^{\grave{\grave{1}}}}2\nabla \text{あ}$$6\sim’arrow 7\triangleright y^{\backslash }\mathrm{i}’\mathrm{A}/\mathrm{J}^{1}\lrcorner 6$$|\underline{\mathrm{B}}\mathrm{b}$,.

$((68250000y-24863426875)x^{2}+(-37182906425y^{2}-10656175824y)x)\partial_{aae}^{2}$$+(-8437500x^{2}+(637980000y+694297170)x-3708061182y^{2})\partial_{y}\partial_{l}$$+((-885281250y-128629930625)x-179156250y^{3}-111958930125y^{2}-54216894564\mathrm{y})\mathrm{a}\mathrm{x}$$+(-18742500x+154288260y)\partial_{y}^{2}$$6$$\mathrm{T}\text{あ}$.

$+(-25312500x-511875000y^{2}-2118711000y+2082891510)\partial_{y}-5545968750y$ -102242340375$\sim-\chi\iota \text{ら}$$\sigma=[$

$q$

)

ffl

$\mathfrak{l}\not\in \mathrm{f}\mathrm{f}\mathrm{f}\mathrm{f}\mathrm{l}\ovalbox{\tt\small REJECT} \text{を}$) $g$$l^{\backslash }\text{て}\sigma \mathit{0}$ae$\text{を_{}\mathrm{p}}^{2}+\mathrm{H}T6\underline{\prime\triangleright}$,$\frac{30517578125}{218041257467152161}\frac{1}{x,3’ 3}-25\frac{1220703125}{1483273860320763,11251},$ $\frac{1}{xy^{2}}+\frac{48828125}{10090298369529,1},\frac{1}{xy^{3}}-\frac{1953125}{68641485507}\frac{1}{xy^{4}}5$

9765625$+ \frac{78125}{466948881}\frac{1}{xy^{5}1}-\overline{3176523}\overline{xy^{6}}\overline{21609}x+--\overline{6\mathrm{f}_{525111}^{7}}\overline{147}\overline{xy^{8}}\overline{1441471195647}^{\frac{1}{x^{2}y}}$$+ \frac{390625}{9805926501}\overline{x^{2}y^{2}}-\frac{15625}{66706983}\frac{1}{x^{2}y^{3}}+\frac{1}{x^{2}y^{4}}-\overline{453789}\overline{3087}\overline{x^{2}y^{5}}+\overline{21}\overline{x^{2}y^{6}}$

3125

1125

15111

$+-\overline{9529569}\overline{x^{3}y}\overline{64827}\overline{xy^{2}}\overline{441}\overline{x^{3}y^{3}}\overline{63}\overline{x^{4}y}+-]$$\text{を}\acute{\tau}\doteqdot 6$

\yen

.

$f_{arrow}^{\vee}$,,$Ann^{(2)}\text{を}\overline{/\mathrm{F}\backslash }\text{点_{}1}\mathrm{t}’.\text{局}\overline{\mathrm{p}}ffi(\mathrm{b}\mathrm{b}f^{\sim}.\mathrm{b}\text{の}[] 2$$\langle y^{8},5y^{4}x+7y^{6},3x^{2}+y^{5}, P_{0}\rangle\underline{\prime\triangleright}r_{X6}$. {F3$\mathrm{b}$,$(620690779589826484980\mathrm{x} +42736086463561823556y^{2})\partial_{l}\partial_{y}$$P_{0}=$$+(32879237237110246500x+16068800153474932500y^{3}-94484544902432603100y^{2}$$+256416518781370941336y)\partial_{l}$$+305257760454013025400y\partial_{y}^{2}$$+((-1315628472217500000y+1439969663151534375)x$$-8814296054882311875y^{2}+103829170222453410000y+3368010623675943713580)\partial_{y}$$-1165771484375000x^{3}+(-2060009765625000y+116844758789062500)x^{2}$$+(-35487436523437500y^{3}+144556613085937500y^{2}$$-473028944554687500y-4697651664385312500)x$$+52404357304687500y^{4}+730337234189062500y^{3}-4874975501945062500y^{2}$$-20807036096166258750y+500110503238150591500$-C$\text{あ}6$. 4$\overline{\tau}7J\triangleright\langle y^{8},5y^{4}x+7y^{6},3x^{2}+y^{5}, P_{0}\rangle \text{の}F$$D_{X}/Ann^{(2)}\text{の_{}\overline{\mathfrak{l}ff\backslash }}\text{点_{}\mathfrak{l}}\mathfrak{l}\acute{\cdot}k^{\backslash }\}\}6\ovalbox{\tt\small REJECT}\Phi \mathrm{E}\hslash^{\mathrm{Y}}>$$\underline{\mathrm{r}}’\supset$.

$f_{\mathit{1}k^{\mathrm{Y}}}$,$\mathrm{f}\mathrm{f}\text{微分}1\not\in \mathrm{f}\mathrm{f}\mathrm{f}\mathrm{f}\mathrm{l}\Leftrightarrow$$P_{1}$,$P_{2}$,$1T\text{あ}$$6_{\sim}-\text{と}$$P\Leftrightarrow \text{の}\mathrm{f}\mathrm{f}\mathrm{l}\mathrm{f}\mathrm{f}\mathrm{i}\backslash \grave{/}\neq|.’\vee\supset\ovalbox{\tt\small REJECT}\backslash \text{て}(\mathrm{g},$&}$\triangleright 7+\mathrm{E}\mathrm{E}\text{を}\#\prod\not\subset 6$$\theta\grave{\grave{\mathrm{l}}}\text{分}\hslash\backslash 6$$\mathrm{X}\mathrm{R}$.

$\{’k’\supset \text{て}$

$[3]$

,

1$!

)

$E_{12}\sigma$$\mathrm{S}$$B_{\mathfrak{o}}^{\mathrm{A}}$$l\iota\gamma_{\wedge}.\iota\backslash$.

$\mathrm{c}\mathrm{k}9$,$+_{J\backslash }\text{ロ}y$$\backslash \backslash \backslash \backslash \backslash yf^{7}\tau*_{\backslash }$, $Ann=Ann^{(2)}$ i $|$\# 2$

157
3$\mathrm{f}\#\Leftrightarrow$

Unimodal, quasihomogeneous, ($ffi|\rfloor\chi_{\vee}\# 4^{1}\text{特異}f_{\mathrm{I}\backslash }5\mathfrak{l}_{\mathrm{X}^{\backslash }}^{-\mathrm{T}\mathrm{b}}.$$\acute{\mathrm{c}^{}}\}1$$\star \text{数}\mathrm{f}\mathrm{f}\mathrm{J}\text{局}\overline{\mathrm{p}}ffi\text{コ_{}J\overline{\backslash }}T\text{モ_{ロ}}$$\underline{k}r_{t\downarrow^{\wedge}2}rx\iota$

2$\ovalbox{\tt\small REJECT} \text{数}$

.

:
:
:
:
:
:
:
:
:
:
:
:
:
:

$E_{12}$$E_{13}$$E_{14}$$Z_{11}$$Z_{12}$$Z_{13}$$W_{12}$$W_{13}$

3$Q_{10}$$\infty_{\mathrm{x}}\text{数}$$\backslash \dot{\nearrow}^{\backslash }-\backslash \text{類を特}\mathrm{f}\mathrm{f}\text{付}\backslash \}\mathrm{e}6\mathrm{f}\text{微分}l\not\in \mathrm{f}\mathrm{f}\mathrm{f}\mathrm{f}\mathrm{l}\ovalbox{\tt\small REJECT}|_{arrow}^{-}\vee\supset \mathrm{t}\backslash \text{て}\ovalbox{\tt\small REJECT}\wedge\cdot 6$$Q_{11}$$Q_{12}$$S_{11}$$S_{12}$$U_{12}$$f(x, y)=x^{3}+y^{7}+axy^{5}$$f(x, y)=x^{3}+xy^{5}+ay^{8}$$f(x, y)=x^{3}+y^{8}+axy^{6}$$f(x, y)=x^{3}y+y^{5}+axy^{4}$$f(x, y)=x^{3}y+xy^{4}+ax^{2}y^{3}$$f(x, y)=x^{3}y+y^{6}+axy^{5}$$f(x, y)=x^{4}+y^{5}+ax^{2}y^{3}$$f(x, y)=x^{4}+xy^{4}+ay^{6}$$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}$$f(x, y, z)=x^{4}+y^{2}z+xz^{2}+ax^{3}z$$f(x, y, z)=x^{2}y+y^{2}z+xz^{3}+az^{5}$$f(x, y, z)=x^{3}+y^{3}+z^{4}+axyz^{2}$, $f_{l}=\partial f(x, y, z)/\partial x$ , $f_{y}=\partial f(x, y, z)/\partial y$,,annihilating ideal. { ,annihilat$(x, y, z)\in X=_{\vee}(-3\mathrm{t}_{arrow \mathrm{X}^{\backslash }}^{-[perp]}1\mathrm{b}$$f_{y}$,$f_{z}\acute{\backslash \cdot\backslash }i\pm R_{\mathrm{c}}^{\mathrm{A}}\text{れ}6$$[1/f_{x}f_{y}f_{z}]U\supset D_{X}$6.
$\backslash /$.

$\ovalbox{\tt\small REJECT}_{\mathit{1}\sim}^{+_{arrow}}$$\sigma x>,$,$\mathit{4}\overline{7}^{-}7\backslash ^{\backslash }J\triangleright I=\langle f_{l}, f_{y}, f_{z}\rangle\}_{arrow \mathrm{X}}^{}\gamma_{\backslash }\mathrm{b}$$\downarrow\sigma)$$\mathcal{F}6$$[1/f_{x}f_{y}f_{z}]\text{を}$$\mathrm{e}$.a$\Pi^{-}-$$f_{z}=\partial f(x, y, z)/\partial z\text{と}$ffl$\iota$$\backslash \xi$$(\S 2\vee\not\in \mathrm{f}\mathrm{f}_{\backslash }1_{\backslash })$.

$k^{\backslash }$$\langle$.

$f_{l}$,$\mathrm{Y}=V(I)\{_{arrow}^{-\underline{\mathrm{A}}}\text{を}3\doteqdot \text{つ}(\star \text{数}\Psi\backslash ]\text{局}\overline{\mathrm{p}}ffi\text{コ_{}J\overline{\backslash }}\triangleleft \text{モロ}\backslash j^{\backslash }\backslash -\text{類類類}$$Ann\underline{\mu}k^{\backslash }\text{く}$$\underline{\mathrm{B}}\mathrm{b}$$D_{X}\dagger 1X\mathrm{k}\sigma)\ovalbox{\tt\small REJECT}\pi_{\acute{\nearrow}\ovalbox{\tt\small REJECT} \text{微分}l\mathrm{E}\mathrm{f}\mathrm{f}\mathrm{f}\mathrm{f}\mathrm{l}\text{素_{}\mathit{0})\ovalbox{\tt\small REJECT}}}’$$\text{と}$-t$\star\backslash j\lceil_{\mathrm{B}}^{\mu \text{の^{}\prime}\ovalbox{\tt\small REJECT}\pi/}//\ovalbox{\tt\small REJECT}\uparrow^{1}d_{JJ}^{1\prime\backslash }\mathfrak{l}\not\in ff\mathrm{f}\mathrm{f}\mathrm{l}\text{素}\mathrm{t}D\xi\in 6\mathrm{R}\mathrm{T}6E\prime \mathrm{r}\overline{7^{-}}7\backslash ^{\backslash }J\triangleright 2:Ann^{(j)}[succeq] \mathrm{k}\supset\backslash$$\overline{|,5^{\exists \mathrm{I}5}\backslash }J\backslash \backslash \{_{arrow\square }^{-arrow\prime}\text{を}\not\in\cdot\doteqdot \text{つ}(\not\leq \text{数}\ovalbox{\tt\small REJECT} \mathrm{j}\backslash \text{局}\overline{\mathrm{p}}ffi^{\text{コ_{}J\overline{\backslash }}}T\text{モ}\mathfrak{s}\supset\backslash i^{\backslash }\backslash -\text{類}[1/f_{x}f_{y}f_{z}]|_{(0,0,0)}[succeq] T6.$$\dagger\ovalbox{\tt\small REJECT} \mathit{0}\supset_{\mathrm{Q}}-\equiv \mathrm{E}\tau- \text{を}\square$. Xffi$\backslash \mu^{\backslash }A\mathrm{T}\text{の}\ovalbox{\tt\small REJECT}.\ovalbox{\tt\small REJECT}\pi_{\acute{\prime}}’|_{arrow^{\vee}\supset \mathrm{V}^{\backslash }\mathrm{C}_{\mathrm{p}}^{\hat{=}}+\ovalbox{\tt\small REJECT}-\tau\epsilon}’\vee$.

$X=\mathbb{C}^{2}\ni(x, y)$(1)@#$\mathrm{b}$@$\mathrm{i}\gamma_{\backslash }\zeta D_{7\backslash \mathfrak{o}}^{\sqrt}+\ovalbox{\tt\small REJECT}_{\backslash }\text{を_{}\acute{\mathrm{f}}^{\mathrm{B}}}\tau f_{-}^{}$

Proposition 1
(i)
(ii)

$-,\mathrm{r}_{\backslash }\text{ロ}\sqrt$$\approx\backslash \backslash /\backslash i-*\tau D_{X}\backslash /Ann^{(1)}\mathrm{t}\mathrm{o}\ovalbox{\tt\small REJECT} \mathrm{g}_{\iota\backslash }\{_{arrow k^{\mathrm{Y}}\downarrow)6\ovalbox{\tt\small REJECT}\ovalbox{\tt\small REJECT} \mathrm{E}\overline{l^{1f}}}^{\vee}\approx=2$$-,\mathrm{r}_{\backslash }\text{ロ}\sqrt$$arrow\backslash \wedge\backslash /\backslash i^{7}\tau\phi_{\backslash }D_{X}/Ann^{(2)}\mathrm{t}D$$\ovalbox{\tt\small REJECT}_{r^{\mathrm{I}}\backslash \backslash }5^{}\acute{\downarrow}-$お t) 6$\ovalbox{\tt\small REJECT} 7E\overline{]^{1t}>}=1$

Theorem 2 $Ann^{(2)}=Ann$

Theorem 3(i)$Homp_{X}(Dx/Ann^{(1)},\mathcal{H}_{[0]}^{\dim X}(Ox))=\mathrm{S}\mathrm{p}\mathrm{a}\mathrm{n}\{\delta_{0}, \sigma\}$$\sim\sim\backslash arrow\acute{.\backslash }$$\acute{\dot{\mathrm{c}}}\text{あ}\xi$

(ii)

4

,
.

.

$\delta 0\mathfrak{l}3;\ovalbox{\tt\small REJECT}_{\backslash J\backslash \backslash }|5’\}_{\square }rightarrowarrow’\text{を}\mathrm{H}$$\text{つ}\overline{\tau}$’ レ $P$$Homp_{X}(D_{X}/Ann^{(2)},\mathcal{H}_{[0]}^{\dim X}(O_{X}))=\mathrm{S}\mathrm{p}\mathrm{a}\mathrm{n}\{\sigma\}$, dirn $X=2$ (D@8}h $[1/xy]$ ).

$\frac{\Xi}{\prime\backslash }\mathrm{f}\mathrm{f}\mathrm{m}_{\backslash }\overline{\overline{\overline{\mathrm{n}}-}}+\mathrm{F}\#\ovalbox{\tt\small REJECT}$’$\vec{\mathrm{B}^{1}\mathrm{J}}\Xi \mathrm{D}\mathrm{t}_{}++|- \text{あ}\downarrow\ovalbox{\tt\small REJECT}_{-}^{\backslash +_{\sim}}\backslash ,\hat{;\mathrm{E}}\text{理_{}\mathrm{t}arrow\ovalbox{\tt\small REJECT}\neq 5T}^{arrow}$ $6_{:}\mathrm{E}\mathfrak{l}\mathrm{K}\mathrm{f}\mathrm{i}0$il$\ovalbox{\tt\small REJECT} \mathrm{f}\mathrm{f}\mathrm{i}\ovalbox{\tt\small REJECT}$$\infty_{\Xi \mathrm{i}\lambda\iota 6^{\frac{}{/I}\grave{)}}}\Xi_{\mathrm{J}}’\backslash$,.

$\mathrm{T}\mathrm{f}\mathrm{f}\mathrm{p}arrow \mathrm{i}\acute{\mathrm{c}^{\grave{\backslash }}}\mathfrak{l}\mathrm{f}a=1\underline{\prime}\mathrm{b}\triangleright\overline{\mathrm{c}}0\Rightarrow+=\ovalbox{\tt\small REJECT} \text{を}J_{-}\uparrow\overline{\tau}\circ f_{arrow}^{\sim}$$x\succ y)\underline{\mu}\mathrm{b}\acute{\mathrm{c}}$,$4\backslash f_{\backslash }\text{数}\not\equiv\not\equiv\ovalbox{\tt\small REJECT} \text{式}|1\lfloor 5_{\backslash }\mathrm{F}\text{を}$$\backslash \}\overline{\mathrm{c}}\mathrm{f}[6]\text{を}\geq,\mathrm{i}\mathrm{B}_{\mathrm{D}_{\backslash }}^{n}\pm.*/|\iota\gamma\sim.\iota\backslash$

.

ffl ’

,

$(\text{コ_{}J}^{-r_{\backslash }\text{モ}\mathrm{C}1} \backslash \nearrow^{\backslash ^{\backslash }}-\backslash \text{類_{}\iota D\#\mathrm{R}\mathrm{f}Ann^{(1)},Ann^{(2)}\mathit{0})\xi\in \mathrm{E}\mathrm{R}\overline{\pi}})\text{を}$$\frac{\backslash \backslash \prime}{*}\downarrow\mathrm{e}^{}6\mathrm{T},,\frac{\mathrm{f}\mathrm{f}\mathrm{i}}{\prime\prime 7-\backslash }\underline{(\not\in}_{7\mathrm{F}’\acute{\mathrm{c}^{\grave{\backslash }}}}’,\text{与_{}\grave{\lambda}\tilde{\mathrm{b}}\text{れ}6\mathrm{U}\mathrm{n}\mathrm{i}\mathrm{m}\circ \mathrm{d}\mathrm{a}1\text{特異_{}l5_{\backslash \backslash }-*_{\backslash }1\mathrm{b}}},\acute{|}_{},$$\text{つ}l$,$\ovalbox{\tt\small REJECT}\neq 7\text{数}(\dim X=3\mathit{0})\pm\ovalbox{\tt\small REJECT}^{\mathrm{B}\bigwedge_{\Pi}}\}\Sigma[1/xyz]$$\not\leq:\mathrm{f}\iota k^{\backslash }\backslash \lambda\iota\ovalbox{\tt\small REJECT}\underline{\prime\triangleright}w\acute{\mathrm{c}}k^{\backslash }$$\langle$$i^{\backslash }.\triangleright 7\grave{\mathcal{F}}\mathrm{E}\Gamma \mathrm{z}\neq \text{の}m\mathfrak{o}=\Rightarrow+\ovalbox{\tt\small REJECT}\dagger \mathrm{f}$.

$\mathrm{T}\mathrm{t}=$$\ovalbox{\tt\small REJECT}\backslash \mathrm{F}\pi//,\}_{arrow}^{arrow}\#\mathrm{J}^{y\backslash ^{\mathrm{o}}}\overline{7}\nearrow-\nearrow-a\emptyset\grave{\grave{>}}$,$x\succ y\succ z(2\ovalbox{\tt\small REJECT} \text{数の}\ovalbox{\tt\small REJECT}_{\mathrm{D}}^{\mathrm{A}}\#\mathrm{J}$$\ovalbox{\tt\small REJECT}$$\overline{\mathrm{c}}\{\overline{\mathrm{T}}0/\gamma_{\wedge}.

\Rightarrow+\mathrm{p}=\ovalbox{\tt\small REJECT}_{\mathrm{F}-ff\mathrm{f}\mathrm{f}\mathrm{l}\ovalbox{\tt\small REJECT}\gamma\sim}’"-\text{数式}k^{\mathrm{h}}\text{理^{}\backslash }\backslash \nearrow\wedge\overline{\tau}\mathrm{A}(\mathrm{k}\mathrm{a}\mathrm{n}/\mathrm{s}\mathrm{m}\mathrm{l}, \mathrm{r}\mathrm{i}\mathrm{s}\mathrm{a}/\mathrm{a}\mathrm{s}\mathrm{i}\mathrm{r})\}_{arrow}’$

158
2Ra

$E_{12}$$E_{13}$$E_{14}$$Z_{11}$$Z_{12}$$Z_{13}$$W_{12}$$W_{13}$$3\overline{\prime R}\mathrm{a}\text{数}$$Q_{10}$$Q_{11}$$Q_{12}$$S_{11}$$S_{12}$$U_{12}$

4.1:$f(x, y)=x^{3}+y^{7}+xy^{5}$

:
:
:
:
:
:
:
:
:
:
:
:
:

$f(x, y)=x^{3}+xy^{5}+y^{8}$$f(x, y)=x^{3}+y^{8}+xy^{6}$$f(x, y)=x^{3}y+y^{5}+xy^{4}$$f(x, y)=x^{3}y$ % $xy^{4}$ %$x^{2}y^{3}$$f(x, y)=x^{3}y+y^{6}+xy^{5}$$f(x, y)=x^{4}+y^{5}+x^{2}y^{3}$$f(x, y)=x^{4}+xy^{4}+y^{6}$$f(x, y, z)=x^{3}+y^{4}+yz^{2}+xy^{3}$$f(x, y, z)=x^{3}+y^{2}z+xz^{3}+z^{5}$$f(x, y, z)=x^{3}+y^{5}+yz^{2}+xy^{4}$$f(x, y, z)=x^{4}+y^{2}z+xz^{2}+x^{3}z$$f(x, y, z)=x^{2}y+y^{2}z+xz^{3}+z^{5}$$f(x, y, z)=x^{3}+y^{3}+z^{4}+xyz^{2}$$E_{12}\mathrm{E}\mathrm{f}\mathrm{f}\mathrm{i}\not\in \mathrm{f}\mathrm{f}\mathrm{i}$$f=x^{3}+y^{7}+xy^{5}$,$f_{v}=3ax^{2}+y^{5},f_{y}=7y^{6}+5xy^{4}$.

Gb $=\{5x^{3}+7y^{2}x^{2},125x^{4}-1029yx^{3},3x^{2}+y^{5}, -21yx^{2}+5y^{4}x\}$.
$I=I_{1}\cap I_{2}$ ,$I_{1}=\langle 25y+147,3125), +151263\rangle$ , $\sqrt{I_{1}}=\langle 25y+147,3125), +151263\rangle$ ,$I_{2}=\langle y^{8},5y^{4}x+7y^{6},3x^{2}+y^{5}\rangle$ ,.
,$P_{1}=$ $(5yx +7y^{3})\partial_{y}+20x$ $+42y^{2}$ ,$\sqrt{I_{2}}=\langle y, x\rangle$$Ann^{(1)}=\langle f_{l}, f_{y}, P_{1}, P_{2}\rangle$$(3500y^{2}+2058y)x\partial_{l}+((-1000y-735)x+7203y^{2})\partial_{y}-4000\mathrm{x}$ $+7700y^{2}+84378y$$P_{2}=$$Ann^{(2)}=\langle f_{l}, f_{y}, P\rangle$.

,$((68250000y-24863426875)x^{2}+(-37182906425y^{2}-10656175824y)x)\partial_{l}^{2}$$P=$$+(-8437500x^{2}+(637980000y+694297170)x-3708061182y^{2})\partial_{y}\partial_{x}$$+((-885281250y-128629930625)x-179156250y^{3}-111958930125y^{2}-54216894564y)\partial_{x}$$+(-18742500x+154288260y)\partial_{y}^{2}$$+(-25312500x-511875000y^{2}-2118711000y+2082891510)\partial_{y}$$-5545968750y$ -102242340375.
$\sigma=[$$-11151 \frac{5^{5}}{3^{3}7^{6}}\frac{1}{5x\mathrm{y}^{6}}+\frac{5^{3}}{3^{2}7^{4}}\frac{1}{xy^{7}}1$$\frac{5^{15}}{3^{8}7^{16}5}\frac{1}{xy}-1\frac{5^{13}}{3^{7}7^{14}5^{1}}\frac{1}{0xy^{2}}+\frac{5^{11}}{3^{6}7^{12}5^{8}}\frac{1}{xy^{3}}-\frac{5^{9}}{3^{5}7^{10}5^{6}}\frac{1}{xy^{4}}+\frac{5^{7}}{3^{4}7^{8}}\frac{1}{xy^{5},4},$$-_{\overline{3\cdot 7^{2}}\overline{xy^{8}}}- \overline{3^{6}7^{11}}\overline{x^{2}y}\overline{3^{5}7^{9}}\overline{x^{2}y^{2}}+-\overline{3^{4}7^{7}}\overline{x^{2}y^{3}}\overline{3^{3}7^{5}}\overline{x^{2}y^{4}}\overline{3^{2}7^{3}}\overline{x^{2}y^{5}}+-+\frac{1}{3\cdot 7}\frac{1}{x^{2}y^{6}}$$+ \frac{5^{5}}{3^{5}7^{6}}\frac{1}{x^{3}y}-\frac{5^{3}}{3^{4}7^{4}}\frac{1}{x^{3}y^{2}}+\frac{5}{3^{3}7^{2}}\frac{1}{x^{3}\oint}-\frac{1}{3^{3}7}\frac{1}{x^{4}y}]$4.2$E_{13}$.

Effiaffi,, $f_{y}=5xy^{4}+8y^{7}$ .

$f=x^{3}+xy^{5}+y^{8}$$f_{a}=3x^{2}+y^{5}$

Gb $=\{4608x^{4}+125yx^{3},3x^{2}+y^{5}, -24y^{2}x^{2}+5y^{4}x, 5x^{3}+8y^{3}\dot{x}^{2}, (-192y^{2}-25y)x^{3}\}$ .
,$I=I_{1}\cap I_{2}$$I_{1}=\langle 192y+25,884736x$ -3125), $\sqrt{I_{1}}=\langle 192y+25,884736x$ -3125),$I_{2}=\langle yx^{3}, x^{4},3x^{2}+y^{5},24y^{2}x^{2}-5y^{4}x, 5x^{3}+8y^{3}x^{2}\rangle$$Ann^{(1)}=\langle f_{l}, f_{y}, P_{1}, P_{2}\rangle$,$\sqrt{I_{2}}=\langle y, x\rangle$.

,$P_{1}=(5x^{2}+8y^{\}x)\partial_{l}+15x$ $+16y^{3}$,$P_{2}=((2688y^{2}+125y)x-360y^{4})\partial_{l}+(-240x+50y^{2})\partial_{y}+5376y^{2}+575y$$Ann^{(2)}=\langle f_{l}, f_{y}, P\rangle$

,

.

159
$((7077888y-4492800)x^{2}+(-13639680y^{3}-139200y^{2}+71875y)x+9000y^{4})\partial_{ae}^{2}$$P=$$+((1244160y+24000)x+28750y^{2})\partial_{y}\partial_{x}$$+((155713536y-3041280)x-43868160 \oint -288000y^{2}+402500y)\partial_{l}$$+(42467328y^{2}+6773760y+24000)\partial_{y}$$+566231040y+55710720$ .

$- \frac{2^{48}3^{8}}{5^{17}}\frac{1}{xy}+\frac{2^{42}3^{7}}{5^{15}}\frac{1}{xy^{2}}-\frac{2^{36}3^{6}}{5^{13}}\frac{1}{xy^{3}}+\frac{2^{30}3^{5}}{5^{11}}\frac{1}{xy^{4}}-\frac{2^{24}3^{4}}{5^{9}}\frac{1}{xy^{5}}+\frac{2^{18}3^{3}}{5^{7}}\frac{1}{xy^{6}}-\frac{2^{12}3^{2}}{5^{5}}\frac{1}{xy^{7}}$$\sigma=[$$2^{6}3$1$2^{33}3^{5}$$1$$2^{9}3$1$+- \frac{1}{5}\frac{1}{xy^{9}}-+\frac{2^{27}3^{4}}{5^{10}}\frac{1}{x^{2}y^{2}}-\frac{2^{21}3^{3}}{5^{8}}\frac{1}{x^{2}y^{3}}+\frac{2^{15}3^{2}}{5^{6}}\frac{1}{x^{2}y^{4}}-\overline{5^{3}}\overline{xy^{8}}\overline{5^{12}}\overline{x^{2}y}\overline{5^{4}}\overline{x^{2}y^{5}}$2359296 112288

1$2^{3}$

1

1$2^{6}$

1.5

1$2^{3}$

1$+-+-\overline{5^{2}}\overline{x^{2}y^{6}}\overline{5^{7}}\overline{x^{3}y}\overline{5^{5}}\overline{x^{3}y^{2}}\overline{5^{3}}\overline{x^{3}\oint}\overline{3}+$$5^{2}\overline{x^{4}y}$$\overline{x^{3}y^{4}}-\overline{3}\cdot$

4.3

].

$E_{14}\mathrm{E}^{1}\mathrm{E}_{\backslash }\mathrm{F}\mathrm{f}_{1}\mathrm{i}$$f=x^{3}+xy^{6}+y^{8}$ ,, $f_{y}=6xy^{5}+8y^{7}$ .
Gb $=\{3x^{3}+4y^{2}x^{2},9yx^{4}-64yx^{3},9x^{5}-64x^{4},3x^{2}+y^{6}, -4yx^{2}+y^{5}x\}$ .
$I=I_{1}\cap I_{2}$ ,$I_{1}=\langle 9x -64,3y^{2}+16\rangle$ , $\sqrt{I_{1}}=\langle 3y^{2}+16,9x-64\rangle$ ,$I_{2}=\langle 3x^{3}+4y^{2}x^{2}, yx^{3}, x^{4},3x^{2}+y^{6},4yx^{2}-y^{5}x\rangle$ ,.
A$f_{x}=3x^{2}+y^{6}$$=\langle y, x\rangle$$Ann^{(1)}=\langle f_{x}, f_{y}, P_{1}, P_{2}\rangle$,$P_{1}=(3x^{2}+4y^{2}x)\partial_{x}+9x$ $+8y^{2}$,$P_{2}=((9y^{3}+32y)x+4y^{5})\partial_{x}+(3y^{4}+16y^{2})\partial_{y}+42y^{3}+176y$$Ann^{(2)}=\langle f_{x}, f_{y}, P\rangle$

$P=$

.

,

$((8064y^{2}+1600512)x^{2}+2076672y^{2}x)\partial_{x}^{2}$$+(-3024yx^{2}+18432yx-4096y^{3})\partial_{y}\partial_{\mathrm{g}}$$+(93312x^{3}+(84807y^{2}+554688)x^{2}+(-7452y^{4}+756288y^{2}+7117824)x$$+20160y^{6}-29184y^{4}+6201344y^{2})\partial_{\mathrm{g}}$$+(-6912x-9216y^{2})\partial_{y}^{2}$$+(31104yx^{2}+(28269y^{3}+268272y)x-2484y^{5}+380736y^{3}-368640y)\partial_{y}$$+435456x^{2}+(354294y^{2}+3050784)x-17172y^{4}+4063680y^{2}+746496$ .
$\sigma=[$$- \frac{3^{4}}{2^{21}1}\frac{1}{xy_{1}}+\frac{3^{3}}{2^{17}}\frac{1}{xy^{3}1}-1\frac{3^{2}}{2^{13}1}\frac{1}{xy^{5}}+\frac{3}{2^{9}}\frac{1}{x_{3^{7}}}-\frac{1}{2^{5}}\frac{1}{xy^{9}}-11\frac{3^{2}}{2^{15}}\frac{1}{x^{2}y}+\frac{3}{2^{11}}\frac{1}{x^{2}y^{3}}-\frac{1}{2^{7}}\frac{1}{x^{2}y^{5}}$$+_{\overline{2^{3}3}\overline{x^{2}y^{7}}}-\overline{2^{9}}\overline{x^{3}y}\overline{2^{5}3}\overline{x^{3}y^{3}}+-\overline{2^{3}3^{2}}\overline{x^{4}y}]$

4.4

.

.

$Z_{11}\ovalbox{\tt\small REJECT}^{\mathrm{J}}\mathrm{f}\# g\mathrm{g}_{1}$$f=x^{3}y+xy^{4}+y^{5}$ ,, $f_{y}=x^{3}+4xy^{3}+5y^{4}$ .
Gb $=\{3yx^{2}+y^{4}, x^{3}-15yx^{2}+4y^{3}x, -11yx^{3}-15y^{2}x^{2},58564x^{5}-37125x^{4}-759375y^{2}x^{2}\}$ .
$f_{x}=3x^{2}y+y^{4}$$I=I_{1}\cap I_{2}$,$I_{1}=$ $(121\mathrm{y}+675,187\mathrm{x} - 10125)$, $\sqrt{I_{1}}=\langle 121y+675,187\mathrm{x}$ -10125),,$I_{2}=\langle 3yx^{2}+y^{4}, x^{3}-15yx^{2}+4y^{3}x, 11yx^{3}+15y^{2}x^{2},11x^{4}+225y^{2}x^{2}\rangle$$Ann^{(1)}=\langle f_{x}, f_{y}, P_{1}, P_{2}\rangle$

$P=$

,

$(270x^{2}+(-792y^{2}-3150y)x -220y^{3})\partial_{x}+(345yx-528y^{3}-2475y^{2})\partial_{y}+1770\mathrm{x}$$Ann^{(2)}=\langle f_{x}, f_{y}, P\rangle$.

,$P_{1}=(33x^{2}+45yx)\partial_{x}+(22yx +30y^{2})\partial_{y}+187x+240y$$P_{2}=$$\sqrt{I_{2}}=\langle y, x\rangle$$-4488y^{2}-18675y$ .

,$(1621488000x^{2}+2211120000yx)\partial_{l}^{2}$$+(-1507537185x^{2}-463117050yx +2171748375y^{2})\partial_{y}\partial_{x}$$+(833546736x^{2}+(-3104767072y-8186767725)x-746016700y^{2}+17492101875y)\partial_{l}$$+(63147480x^{2}-1332723150yx -1934772750y^{2})\partial_{y}^{2}$$+((555697824y-10098876810)x-2041779168y^{2}-29844557550y)\partial_{y}$4-4723431504x–l7355l22928y-112679770725

160$\sigma=[$$\frac{11^{11}}{3^{17}5^{12}}\frac{1}{xy}-\frac{11^{9}}{3^{14}5^{10}}\frac{1}{xy^{2}}+\frac{11^{7}}{3^{11}5^{8}}\frac{1}{xy^{3}}-\frac{11^{5}}{3^{8}5^{6}}\frac{1}{xy^{4}}+\frac{11^{3}}{3^{5}5^{4}}\frac{1}{xy^{5}}-\frac{11}{3^{2}5^{2}}\frac{1}{xy^{6}}+\frac{11^{8}}{3^{13}5^{9}}\frac{1}{x^{2}y}-\frac{11^{6}}{3^{10}5^{7}}\frac{1}{x^{2}y}$$+ \frac{11^{4}}{3^{7}5^{5}}\frac{1}{x^{2}y^{3}}-\frac{11^{2}}{3^{4}5^{3}}\frac{1}{x^{2}y^{4}}+\frac{1}{3\cdot 5}\frac{1}{x^{2}y^{5}}+\frac{11^{5}}{3^{9}5^{6}}\frac{1}{x^{3}y}-\frac{11^{3}}{3^{6}5^{4}}\frac{1}{x^{3}y^{2}}+\frac{11}{3^{3}5^{2}}\frac{1}{x^{3}y^{3}}$$+ \frac{11^{2}}{3^{6}5^{3}}\frac{1}{x^{4}y}-\frac{1}{3^{2}5}\frac{1}{x^{4}y^{2}}-\frac{1}{3}\frac{1}{x^{5}y}]$4.5.

$Z_{12}\ovalbox{\tt\small REJECT} \mathrm{f}\mathrm{l}\not\in \mathrm{f}.\mathrm{i}$$f=x^{3}y+xy^{4}+x^{2}y^{3}$,$f1=3x^{2}y+y^{4}+2xy^{3}$ , $f_{2}=x^{3}+4xy^{3}+3x^{2}y^{2}$.

Gb $=\{3yx^{2}+y^{4}+2y^{3}x$ , $x^{3}+(3y^{2}-6y)x^{2}-2y^{4}$ , $(-14y+5)x^{3}-30yx^{2}-5y^{5}-10y^{4}$ ,$(105y-55)x^{4}+484yx^{3},735x^{5}+605x^{4}-5324yx^{3}\}$,$I=I_{1}\cap I_{2}$$I_{1}=\langle 35y-121,245x+1331\rangle$.

, $\sqrt{I_{1}}=\langle 35y-121,245x +1331\rangle$ ,$I_{2}=\langle 3yx^{2}+y^{4}+2y^{3}x, x^{3}+(3y^{2}-6y)x^{2}-2y^{4},5x^{4}-44yx^{3}, (14y-5)x^{3}+30yx^{2}+5y^{5}+10y^{4}\rangle$$\sqrt{I_{2}}=\langle y, x\rangle$.

$Ann^{(1)}=\langle f_{l}, f_{y}, P_{1}, P_{2}\rangle$,$P_{1}=$$(140x^{2}+248yx+44y^{2})\partial_{l}+((70y+33)x+125y^{2})\partial_{y}+770\mathrm{x}$ $+1158y$$P_{2}=$$((84y+124)x^{2}+(-672y^{2}+2770y)x-42y^{3}-176y^{2})\partial_{ae}$,$+((42y^{2}-169y-132)x-609y^{3}+2008y^{2})\partial_{y}+(462y+340)x-4914y^{2}+16686y$$Ann^{(2)}=\langle f_{l}, f_{y}, P\rangle$

,

.

,

$(194685857871120x^{2}+420775133985360yx +180463194172800y^{2})\partial_{l}^{2}$$P=$$+(-82832632888320x^{2}+(-9751678231680y-180463194172800)x+107193098960640y^{2})\partial_{y}\partial_{x}$$+(-133092828502732x^{3}+(1373829024443004y-3548483873971782)x^{2}$$+(-315326509783170y^{2}+2838895228228440y+76535249452200)x$$-696421142828840y^{3}-81448893497400y^{2}+1647347302339200y)\partial_{l}$$+((-92078741076840y+29963270244600)x-70175382520320y^{2}-210540393201600y)\partial_{y}^{2}$$+((-66546414251366y+350146787028213)x^{2}$$+(1071890730792481y^{2}-3093414693538428y-755979214347180)x$$-325846150902008y^{3}+2066998836578100y^{2}-1513707145196400y-1864786339785600)\partial_{y}$$-732010556765026x^{2}+(7765120865745538y-20427450896819658)x$$-3303190350044904y^{2}+17240073586133520y$ -6930858351546360.
$\sigma=[$$\frac{5^{5}7^{7}}{11^{1\S}5^{4}}\frac{1}{xy,5},+\frac{5^{4}7^{6}}{11^{11}5},\frac{1}{x,7?^{2}}+\frac{5^{3}7^{5}}{11^{9}}\frac{1}{xy^{3},273’}+\frac{5^{2}7^{4}}{11^{7}}\frac{1}{xy^{4},5\cdot 7},+\frac{5\cdot 7^{3}}{111^{5}}\frac{1}{xy^{5}7}+\frac{7^{2}}{1113}\frac{1}{xy^{6}}-\frac{3}{11}\frac{1}{xy^{7}}7131512$$-\overline{11^{10}}\overline{x^{2}y}-\overline{11^{8}}\overline{x^{2}y^{2}}-\overline{11^{6}}\overline{x^{2}\oint}-\overline{11^{4}}\overline{x^{2}y^{4}}-\overline{11^{2}}\overline{x^{2}y^{5}}$$5^{3}7^{3}$$5^{2}7^{2}$

$1$

1

5

$\cdot 7$1

5$5^{2}71$1.

$++++ \frac{1}{11}\frac{1}{x^{3}y^{4}}---\frac{2^{2}}{11}\frac{1}{x^{5}y}\overline{11^{7}}\overline{x^{3}y}\overline{11^{5}}\overline{x^{3}y^{2}}\overline{11^{3}}\overline{x^{3}y^{3}}\overline{11^{4}}\overline{x^{4}y}\overline{11^{2}}\overline{x^{4}y^{2}}]$

4.6$Z_{13}\mathrm{E}\mathrm{f}\mathrm{f}\mathrm{i}\not\in\hslash$$f=x^{3}y+y^{6}+xy^{5}$,,.
Gb $=\{-7yx^{3}-9y^{2}x^{2},3yx^{2}+y^{5}, x^{3}-18yx^{2}+5y^{4}x, -343x^{5}-729y^{3}x^{2}\}$ .
$f_{x}=3x^{2}y+y^{5}$$I=I_{1}\cap I_{2}$$f_{2}=x^{3}+6y^{5}+5xy^{4}$,$I_{1}=\langle 7x +9y, 49y^{2}+243\rangle$, $\sqrt{I_{1}}=\langle 49y^{2}+243,7x +9y\rangle$ ,$I_{2}=(7yx^{3}+9y^{2}x^{2},7x^{4}+162y^{2}x^{2},3yx^{2}+y^{5},$ $x^{3}-18yx^{2}+5y^{4}x$$Ann^{(1)}=\langle f_{l}, f_{y}, P_{1}, P_{2}\rangle$,$P_{1}=(14x^{2}+18yx)\partial_{l}+(7yx +9y^{2})\partial_{y}+77x$ $+90y$,$y^{3}x^{2}\rangle$,$\sqrt{I_{2}}=\langle y, x\rangle$,$P_{2}=(72x^{2}+(-280y^{3}-891y)x-105y^{4})\partial_{l}+(99yx -140y^{4}-567y^{2})\partial_{y}+486x$$Ann^{(2)}=\langle f_{l}, f_{y}, P\rangle$,.

-1540y$-5184\mathrm{y}$

161
$P=$

$(3662593200x^{2}+4709048400yx)\partial_{x}^{2}$$+(-2683171008x^{2}+1890641088yx +6866270208y^{2})\partial_{y}\partial_{ae}$$+(-42710470250x^{3}-105737060100yx^{2}+(-97769239650y^{2}-90813857211)x$$-7368256665y^{3}+55324766448y)\partial_{aae}$$+(262919790x^{2}-3426024924y\mathrm{x} -4839511698y^{2})\partial_{y}^{2}$$+(-21355235125yx^{2}+(-52868530050y^{2}-19843900587)x-48884619825y^{3}$ -1076714$-234907586375x^{2}-554097099675yx$ $-502429306950y^{2}$ -667382260608.
$\sigma=[$$+ \frac{7^{4}}{2\cdot 3^{12}}\frac{1}{x^{2}y^{2}}-\frac{7^{2}}{2\cdot 3^{7}}\frac{1}{x^{2}y^{4}}+\frac{1}{2\cdot 3^{2}}\frac{1}{x^{2}y^{6}}11$$\frac{7^{7}}{2\cdot 3^{19}7^{3}}\frac{1}{xy}-\frac{7^{5}}{2\cdot 3^{14}}\frac{1}{xy^{3}}+\frac{7^{3}}{2\cdot 3^{9}}\frac{1}{xy^{5}}-\frac{7}{2\cdot 3^{4}1}\frac{1}{xy^{7},1},$$-+ \frac{7}{2\cdot 3^{5}}-\frac{1}{2\cdot 33}\frac{1}{x^{4}y^{2}}-\overline{2\cdot 3^{10}}\overline{x^{3}y}\overline{x^{3}y^{3}}\overline{3}\overline{x^{5}y}]$4.7.

$W_{12}\underline{\mathrm{R}^{1}}\mathrm{E}_{\backslash }\Leftrightarrow \mathrm{f}\mathrm{f}\mathrm{i}$,, $f_{y}=5y^{4}+3x^{2}y^{2}$ .

$f=x^{4}+y^{5}+x^{2}y^{3}$$f_{x}=4x^{3}+2xy^{4}$

Gb $=\{3y^{2}x^{2}+5y^{4},2x^{3}+y^{4}x, -9x^{5}-50x^{3},6x^{4}-5y^{6}\}$ .
,$I_{1}=\langle 3y-10,27x^{2}+500\rangle$ , $\sqrt{I_{1}}=\langle 3y-10,27x^{2}+500\rangle$ ,$I_{2}=\langle 2x^{3}+y^{3}x, 3y^{2}x^{2}+5y^{4}, yx^{3},6x^{4}-5y^{5}, x^{5}\rangle$ ,$I=I_{1}\cap I_{2}$$\sqrt{I_{2}}=\langle y, x\rangle$.

$P_{1}=((12y-30)x^{2}+5y^{3})\partial_{\mathrm{g}}+(9y^{2}-30y)x\partial_{y}+(78y-210)x$,$Ann^{(1)}=\langle f_{x}, f_{y}, P_{1}, P_{2}\rangle$,.

$P_{2}=(12x^{3}+(5y^{2}+50y)x)\partial_{l}+(9yx^{2}+50y^{2})\partial_{y}+78x^{2}+350y$$Ann^{(2)}=\langle f_{x}, f_{y}, P\rangle$

$P=$

,

$(12000x^{2}+20000y^{2})\partial_{l}^{2}+(144000y-480000)x\partial_{y}\partial_{x}$$+(-291519x^{3}+(-81675y^{2}-1009800y-933000)x)\partial_{x}$$+(-45000x^{2}+69000y^{Z}-480000y)\partial_{y}^{2}$$+((-210681y+106110)x^{2}-723600y^{2}+240000y-3840000)\partial_{y}$$-1878957x^{2}-5788800y$ -5499000.
$\sigma=[$$- \frac{3^{6}}{10^{7}}\frac{1}{xy}-\frac{2^{5}}{10^{6}}\frac{1}{xy^{2}}-\frac{3^{4}}{10^{5}}\frac{1}{xy^{3}}-\frac{3^{3}}{10^{4}}\frac{1}{xy^{4}}-\frac{3^{2}}{10^{3}}\frac{1}{xy^{5}}-\frac{3}{10^{2}}\frac{1}{xy^{6}}$$+ \frac{3^{3}}{2\cdot 10^{4}}\frac{1}{x^{3}y}+\frac{3^{2}}{2\cdot 10^{3}}\frac{1}{x^{3}y^{2}}+\frac{3}{2\cdot 10^{2}}\frac{1}{x^{3}y^{3}}+\frac{1}{2\cdot 10}\frac{1}{x^{3}y^{4}}-\frac{1}{2^{2}10}\frac{1}{x^{5}y}]$4.8.

$W_{13}\underline{\Phi^{\mathrm{J}}}\#\Leftrightarrow,\mathrm{f}_{1}.1_{\backslash }$$f=x^{4}+xy^{4}+y^{6}$ ,$f_{x}=4x^{3}+y^{4}$Gb, $f_{y}=4xy^{3}+6y^{5}$ .

$=\{4x^{3}+y^{4}, -6yx^{3}+y^{3}x, -9x^{6}-x^{5}\}$$I=I_{1}\cap I_{2}$.

,$I_{1}=\langle 9x+1,27y^{2}-2\rangle$, $\sqrt{I_{1}}=\langle 27y^{2}-2,9x +1\rangle$ ,,.

$I_{2}=\langle 4x^{3}+y^{4},6yx^{3}-y^{3}x, y^{3}x^{2}, x^{5}\rangle$$Ann^{(1)}=\langle f_{\mathrm{g}}, f_{y}, P_{1}, P_{2}\rangle$$\sqrt{I_{2}}=\langle y, x\rangle$,$P_{1}=(2x^{2}+3y^{2}x)\partial_{x}+8x$ $+9y^{2}$,$P_{2}=$ $(270yx^{2}+8yx -33y^{3})\partial_{l}+(-36x^{2}+6y^{2})\partial_{y}+810y\mathrm{x}$$Ann^{(2)}=\langle f_{x}, f_{y}, P\rangle$$P=$$+50y$.

,$(4032x^{2}+(-5778y^{2}+896)x+2673y^{4}-168y^{2})\partial_{l}^{2}+(4512yx -2304y^{3}+672y)\partial_{l}\partial_{y}$$+((192456y^{2}+34272)x-31068y^{2}+6496)\partial_{l}+(336x +504y^{2})\partial_{y}^{2}+(96228y^{3}-600\mathrm{y})\mathrm{d}\mathrm{s}$$+1058508y^{2}$$\sigma=[$–22680.

$- \frac{3^{9}}{2^{5}}\frac{1}{xy}-\frac{3^{6}}{2^{4}}\frac{1}{xy^{3}}-\frac{3^{3}}{23}\frac{1}{xy^{5}}-\frac{1}{2^{2}}\frac{1}{xy^{7}}+\frac{3^{7}}{25}\frac{1}{x^{2}y}+\frac{3^{4}}{2^{4}}\frac{1}{x^{2}y^{3}}+\frac{3}{23}\frac{1}{x^{2}y^{5}}-\frac{3^{5}}{2^{5}}\frac{1}{x^{3}y}$$3^{2}$

1$3^{3}$

1

1

1

3

1

$-++-\overline{2^{4}}\overline{x^{3}y^{3}}\overline{2^{5}}\overline{x^{4}y}\overline{2^{4}}\overline{x^{4}y^{3}}\overline{2^{5}}\overline{x^{5}y}]$

.

162
4.9

EfiLE

$Q_{10}$$f=x^{3}+y^{4}+yz^{2}+xy^{3}$,.

$\mathrm{G}\mathrm{b}=\{zy, z^{3},3x^{\int}+y^{3},-12x^{2}+3y^{2}x+z^{2},zx^{2}, -3x^{3}-4yx^{2}\}f_{l}=3x^{2}+y^{3},f=4y^{3}+z^{2}+3xy^{2},f_{z}=2yz.$$I=I_{1}\cap I_{2}$,$I_{1}=$ $(\mathrm{z}\mathrm{y}, z^{3}, zx^{2},3x^{2}+y^{3}, 12x^{2}-3y^{2}x-z^{2},16yx^{2}+z^{2}x, 12x^{3}-z^{2}x)$$I_{2}=\langle z, 3y+16,9x -64\rangle$, $\sqrt{I_{2}}=\langle z, 3y+16,9x -64\rangle$ .
,,,$\sqrt{I_{1}}=\langle z, y, x\rangle$,$Ann^{(1)}=\langle f_{l}, f_{y}, f_{z}, P_{1}, P_{2}, P_{3}\rangle$$P_{1}=$$2zx\partial_{l}+2zy\partial_{y}+3z^{2}\partial_{z}+15z$$P_{2}=$$(18x^{2}+24yx)\partial_{l}+(12yx +16y^{2})\partial_{y}+(21zx +24zy)\partial_{z}+111\mathrm{x}$$P\epsilon$$+136y$ ,$(162x^{3}+(72y-1152)x^{2}-512yx -32z^{2})\partial_{l}+(108yx^{2}-768yx)\partial_{y}+(189zx^{2}-1344zx)\partial_{z}$$=$$+999x^{2}+(144y-7104)x+192y^{2}-1024y$$Ann^{(2)}=\langle f_{l}, f_{y}, f_{z}, P\rangle$

$P=$

.

,

$(1536\mathrm{x} -384y^{2})\partial_{l}^{2}+(768\mathrm{x} +1024y)\partial_{y}\partial_{l}+1536z\partial_{z}\partial_{\mathrm{g}}+(-1458x-1080y+10240)\partial_{l}$$+(432x+486y^{2}+3168y)\partial_{y}^{2}+4224z\partial_{z}\partial_{y}+(3888y+21696)\partial_{y}$$+(1296x^{2}-2304y^{2})\partial_{z}^{2}+729z\partial_{z}+2187$$\sigma=[$.

$- \frac{3^{4}}{2^{21}1}\frac{1}{xt^{Z}}+\frac{3^{3}}{2^{17}}\frac{1}{xy^{2}z}-\frac{3^{2}}{2^{13}}\frac{1}{xy^{3_{Z}}1}+\frac{3}{2^{9}}\frac{1}{1xy^{4_{Z}}}-\frac{1}{2^{5}}\frac{1}{xy^{5_{Z}}1}-\frac{3^{2}}{2^{15}}\frac{1}{x^{2}yz}-\frac{1}{2\cdot 3}\frac{1}{x^{2}yz^{3}}+\frac{3}{2^{11}}\frac{1}{x^{2}y^{2}z}1$$-+ \frac{1}{2^{3}3}\frac{1}{x^{2}y^{4_{Z}}}-+\frac{1}{x^{3}y^{2_{Z}}}-\frac{1}{x^{4}yz}]\overline{2^{7}}\overline{x^{2}y^{3}z}\overline{2^{9}}\overline{x^{3}yz}\overline{2^{5}3}\overline{2^{3}3^{2}}$

4.10

.

$Q_{11}\mathrm{E}\mathrm{f}\mathrm{f}\not\in \mathrm{f}\mathrm{f}\mathrm{i}$$f=x^{3}+y^{2}z+xz^{3}+z^{5}$$f_{l}=3x^{2}+z^{3}$Gb, $f_{y}=2yz$ , $f_{z}=3xz^{2}+y^{2}+5z^{4}$ .

$=\{zy, 3x^{2}+z^{3}, y^{3}, -15zx^{2}+3z^{2}x+y^{2}, yx^{2},375x^{4}+9x^{3}-5y^{2}x\}$$I=I_{1}\cap I_{2}$,$I_{1}=\langle 25z+3, y, 125x +3\rangle$.

, $\sqrt{I_{1}}=\langle 25z+3, y, 125x +3\rangle$ ,$I_{2}=\langle zy, 3x^{2}+z^{3},15zx^{2}-3z^{2}x-y^{2}, y^{3}, yx^{2},9x^{3}-5y^{2}x\rangle$$Ann^{(1)}=\langle f_{l}, f_{y}, f_{z}, P_{1}, P_{2}, P_{3}\rangle$,$\sqrt{I_{2}}=\langle z, y, x\rangle$.

,$P_{1}=$ $(120x^{2}-18zx +10z^{3})\partial_{l}+(120yx -21zy)\partial_{y}+(60zx -12z^{2})\partial_{z}+660\mathrm{x}$-lllz,$P_{2}=((400z^{2}+18z)x-50z^{3})\partial_{l}+21zy\partial_{y}+(-60zx+12z^{2})\partial_{z}-60x$ $+800z^{2}+111z$$P\epsilon$

$=$

,
.

$((400z+18)yx -50z^{2}y)\partial_{l}+(-180zx^{2}-300z^{3_{X}}+21y^{2})\partial_{y}+(-60yx+12yz)\partial_{z}+111y$$Ann^{(2)}=\langle f_{ae}, f_{y}, f_{z}, P\rangle$

$P=$

,

$(162x +270z^{2})\partial_{z}\partial_{l}+(-1500x+840z)\partial_{x}+(1425z^{2}+171z)x\partial_{y}^{2}+246y\partial_{z}\partial_{y}+750y\partial_{y}$.

$+(285x+1375z^{2}+108z)\partial_{z}^{2}+(10000z+1164)\partial_{z}+6500$$\sigma=[$$\frac{5^{10}}{2\cdot 3^{6}}\frac{1}{xyz}-\frac{5^{8}}{2\cdot 3^{5}}\frac{1}{xyz^{2}}+\frac{5^{6}}{2\cdot 3^{4}}\frac{1}{5XP^{z^{3}}}-\frac{5^{4}}{2\cdot 3^{3}}\frac{1}{xyz^{4}}+\frac{5^{2}}{2\cdot 3^{2}}\frac{1}{xyz^{5}}-1\frac{1}{2\cdot 3}\frac{1}{xyz^{6}}$$- \frac{5^{7}}{2\cdot 3^{5}}\frac{1}{x^{2}yz}+\frac{5^{5}}{2\cdot 3^{4}}\frac{1}{x^{2}yz^{2}}-\overline{2\cdot 3^{3}}\overline{x^{2}yz^{3}}+\frac{5}{2\cdot 3^{2}}\frac{1}{x^{2}yz^{4}}-\frac{1}{2\cdot 3}\frac{1}{x^{2}y^{\theta_{Z}}}$$+ \frac{5^{4}}{162}\frac{1}{x^{3}yz}-\frac{5^{2}}{2\cdot 3^{3}}\frac{1}{x^{3}yz^{2}}+\frac{1}{2\cdot 3^{2}}\frac{1}{x^{3}yz^{3}}-\frac{5}{2\cdot 3^{3}}\frac{1}{x^{4}yz}]$41$Q_{12}$.

Effififfi$f=x^{3}+y^{5}+yz^{2}+xy^{4}$,.

$\mathrm{G}\mathrm{b}=\{zy, z^{3}, zx^{f},-4x^{3}-5yx^{2},3x^{2}+y^{4}, -15x^{2}+4y^{3}x+z^{2}\}f_{l}=3x^{2}+y^{4},f=5y^{4}+z^{2}+4xy^{3},f_{z}=2yz$$I=I_{1}\cap I_{2}$,$I_{1}=\langle zy, z^{3}, zx^{2},75yx^{2}+4z^{2}x, 15x^{3}-z^{2}x, 3x^{2}+y^{4},15x^{2}-4y^{3}x-z^{2}\rangle$$I_{2}=\langle z,4x +5y, 16y^{2}+75\rangle$,$Ann^{(1)}=\langle f_{l}, f_{y}, f_{z}, P_{1}, P_{2}, P_{3}\rangle$$\sqrt{I_{2}}=\langle z, 4x +5y, 16y^{2}+75\rangle$

,

.

,

$\sqrt{I_{1}}=\langle z, y, x\rangle$

,

163
,

$P_{1}=$$zx\partial_{\mathrm{g}}+zy\partial_{y}+2z^{2}\partial_{z}+9z$$P_{2}=$$(8x^{2}+10yx)\partial_{ae}+(4yx +5y^{2})\partial_{y}+(10zx +10zy)\partial_{z}+50\mathrm{x}$ $+55y$$P_{3}=$$((192y^{2}+600)x^{2}-375yx -20z^{2})\partial_{l}+(-120x^{2}+96y^{3}x-375y^{2})\partial_{y}$,.

$+(240zy^{2}x-40zy^{3}-750zy)\partial_{z}+(1200y^{2}+1200)x-3375y$$Ann^{(2)}=\langle f_{x}, f_{y}, f_{z}, P\rangle$

$P=$

,

$(3750\mathrm{x} -1000y^{3})\partial_{ae}^{2}+(1500\mathrm{x} +1875y)\partial_{y}\partial_{x}+3750z\partial_{z}\partial_{l}+(-4608yx-3360y^{2}+24375)\partial_{l}$$+(720x+1152y^{3}+6300y)\partial_{y}^{2}+11850z\partial_{z}\partial_{y}+(12672y^{2}+52950)\partial_{y}$.

$+(2880yx^{2}-4500y^{3})\partial_{z}^{2}+13824y$$\sigma=[$$- \frac{2^{9}}{3^{3}5^{6}}\frac{1}{xy^{2_{Z}}}+\frac{2^{5}}{3^{2}5^{4}}\frac{1}{xy^{4}z,2},$$- \frac{2}{3\cdot 5^{2}}\frac{1}{xq^{6_{Z}}}+\frac{2^{7}}{3^{3}5^{5}}\frac{1}{x^{2}yz}-\frac{1}{2\cdot 3}\frac{1}{x^{2}yz^{3}}-\frac{2^{3}}{3^{2}5^{3}}\frac{1}{x^{2}y^{3_{Z}}}$$+ \frac{1}{2\cdot 3\cdot 5}\frac{1}{x^{2}y^{5_{Z}}}+\frac{1}{x^{3}y^{2_{Z}}}-\overline{3^{2}5^{2}}\overline{2\cdot 3^{2}5}^{\frac{1}{x^{4}yz}]}$

4.12

.

$S_{11}\ovalbox{\tt\small REJECT} \mathrm{E}\mathrm{F}\mathrm{k}$$f=x^{4}+y^{2}z+xz^{2}+x^{3}z$ ,, $f_{y}=2yz$ , $f_{z}=y^{2}+2xz+x^{3}$ .
$-4y^{2}+z^{2}$ , $x^{3}+2zx$$(12y^{2}+15z^{2}+64z)x+32y^{2}-8z^{2},3zx^{2}-8zx$,
Gb$-512z^{2}x+75z^{4}+64z^{3}$ , $(-15z^{3}-64z^{2})x+8z^{3}\}$ .
$I=I_{1}\cap I_{2}$ ,$5x$$I_{1}=(25\mathrm{z}+128,$, $\sqrt{I_{1}}=\langle 25z+128, y, 5x-16\rangle$ ,$f_{x}=4x^{3}+z^{2}+3x^{2}z$$=$$\{zy$,$y^{3}$$+y^{2}$,$-16\rangle$

$y,$

,

$I_{2}=\langle zy, 8z^{2}x-z^{3},3zx^{2}-8zx -4y^{2}+z^{2}, y^{3}, (96y^{2}+512z)x+256y^{2}+15z^{3}-64z^{2}, x^{3}+2zx +y^{2}, z^{4}\rangle$$\sqrt{I_{2}}=\langle z, y, x\rangle$.

$Ann^{(1)}=\langle f_{x}, f_{y}, f_{z}, P_{1}, P_{2}, P_{3}\rangle$,,$P_{1}=(4\mathrm{y}\mathrm{x}+2\mathrm{z}\mathrm{y})\mathrm{a}\mathrm{x}+4y^{2}\partial_{y}+3zyx\partial_{z}+3yx+36y$$P_{2}=$$(64x^{2}+56zx +10z^{2})\partial_{x}+(-64xy+20zy)\partial_{y}+(24zx^{2}+15z^{2}x+96y^{2})\partial_{z}+24x^{2}+15zx$ $+84z$,$P_{3}=(16xz+10z^{2})\partial_{x}+(-96xy -16yz)\partial_{y}+(15xz^{2}+96y^{2}-48z^{2})\partial_{z}+(15z-288)x-168z$ .

,$Ann^{(2)}=\langle f_{x}, f_{y}, f_{z}, P\rangle$$P=$$(480\mathrm{z} +300z)\partial_{x}^{2}-288y\partial_{x}\partial_{y}+(-2048x-1280z)\partial_{z}+(-144x-90z-608)\partial_{x}$$(-360\mathrm{z}\mathrm{x}+1152z)\partial_{y}^{2}-2560y\partial_{y}\partial_{z}+144y\partial_{y}+(-2304x^{2}-1344zx-540z^{2}-3072z)\partial_{z}^{2}$$+((-135z-2112)\mathrm{s}-3168z-19968)\partial_{z}-135\mathrm{x}$$\sigma=[$–2088.

$- \frac{5^{5}}{2^{23}}\frac{1}{xyz}+\frac{5^{3}}{2^{16}}\frac{1}{xyz^{2}}-\frac{5}{2^{9}}\frac{1}{xyz^{3}}-\frac{1}{2^{2}}\frac{1}{xyz^{4}}-\frac{5^{4}}{2^{19}}\frac{1}{x^{2}yz}+\frac{5^{2}}{2^{12}}\frac{1}{x^{2}yz^{2}}-\frac{1}{2^{5}}\frac{1}{x^{2}yz^{2}}-\frac{5^{3}}{2^{15}}\frac{1}{x^{3}y^{3_{Z}}}$

5

1

1

1

$5^{2}$

1

1

5

1

1

.

$+_{\overline{2^{8}}\overline{x^{3}yz^{2}}}-\overline{2^{3}}\overline{x^{3}y^{3_{Z}}}-\overline{2^{11}}\overline{x^{4}yz}\overline{2^{4}}\overline{x^{4}yz^{2}}\overline{2^{7}}\overline{x^{5}yz}+-]$

4.13$S_{12} \sum^{/}\# F\mathrm{f}.\mathrm{i}$$f=x^{2}y+y^{2}z+xz^{3}+z^{5}$ ,$f_{x}=2xy+z^{3}$ , $f_{y}=x^{2}+2yz$ , $f_{z}=y^{2}+3xz^{2}+5z^{4}$.

, $(-10zy+3z^{2})x+y^{2},$ $-13y^{2}x-20z^{2}y^{2},$ $-169y^{2}x$ $-800zy^{3},2197y^{2}x+32000y^{4}$ ,$-4000y^{3}x+130y^{3}+507zy^{2}\}$ .
$I=I_{1}\cap I_{2}$ ,$I_{1}=\langle 40y-13z, 400x-16), 8000z^{2}+2197\rangle$ , $\sqrt{I_{1}}=\langle 8000z^{2}+2197,40y-13z, 400x-169\rangle$ ,$I_{2}=\langle x^{2}+2zyz, 2yx+z^{3}, (10zy-3z^{2})x-y^{2},10y^{3}+39zy^{2}, y^{2}x, z^{2}y^{2}\rangle$ ,.
$\mathrm{G}\mathrm{b}=$,$\{x^{2}+2zy$ $2yx$ $+z^{3}$$\sqrt{I_{2}}=\langle z, y, x\rangle$$Ann^{(1)}=\langle f_{x}, f_{y}, f_{z}, P_{1}, P_{2}\rangle$$P_{1}=$,$(104zx +1800z^{2}y-425z^{3})\partial_{x}+((600zy-295z^{2})x+130zy)\partial_{y}+(-240zy+78z^{2})\partial_{z}$$-300zx$ $-30y+3\mathrm{i}2\mathrm{z}$ ,$P_{2}=$$((17000zy-3000z^{2})x-1100y^{2}-7605zy+1521z^{2})\partial_{x}$$+((3000y-3000z)x^{2}+(-2535y+1521z)x-750z^{2}y)\partial_{y}$$+(-2400yx +1014y)\partial_{z}+1170x+11000zy+1950z^{2}$ .
$Ann^{(2)}=\langle f_{x}, f_{y}, f_{z}, P\rangle$

,

164
$P=$

$(4732x +7280z^{2})\partial_{l}^{2}+(-4550zx+5915y)\partial_{l}\partial_{y}+(-28700zx+26390y+3549z)\partial_{l}\partial_{z}$$+(904300x+1500000z^{2}+36673)\partial_{ae}+(-12600zy+4095z^{2})\partial_{y}\partial_{z}$$+((-9000000y+2055000z)x+1244400y-37310z)\partial_{y}+(9555x -90000zy+43950z^{2})\partial_{z}^{2}$$+(-3000000zx+1020000y+1012000z)\partial_{z}-21000000\mathrm{x}$ +6916600.
$\sigma=[$$- \frac{2^{13}5^{6}}{13^{7}}\frac{1}{xyz^{2}}+\frac{2^{7}5^{3}}{13^{4}}\frac{1}{xyz^{4}}-\frac{2}{13}\frac{1}{xyz^{6}}-\frac{2^{10}5^{5}}{13^{6}}\frac{1}{xy^{2}z}+\frac{2^{4}5^{2}}{13^{3}}\frac{1}{xy^{2}z^{3}}+\frac{2\cdot 5}{13^{2}}\frac{1}{xy^{3}z^{2}}-\frac{3}{13}\frac{1}{xy^{4}z}$1$2^{9}5^{4}$

1$2^{3}5$1$2^{6}5^{3}$$2^{5}5^{2}$

1$2^{2}5$1

2

1$-+ \overline{13^{5}}\overline{x^{2}yz^{2}}\overline{13^{2}}\overline{x^{2}yz^{2}}-\overline{13^{4}}\overline{x^{2}y^{2}z}+\frac{1}{13}\frac{1}{x^{2}y^{2}z^{3}}-\overline{13^{3}}\overline{x^{3}yz^{2}}--\overline{13^{2}}\overline{x^{3}y^{2_{Z}}}\overline{13}\overline{x^{4}yz^{2}}]$

4.14$U_{12}$.

型特異点,, $f_{y}=3y^{2}+xz^{2}$ , $f_{z}=4z^{3}+2xyz2$$f=x^{3}+y^{3}+z^{4}+xyz^{2}$$f_{l}=3x^{2}+yz^{2}$

Gb $=\{3x^{2}+z^{2}y, z^{2}x+3y^{2}, zyx +2z^{3}, -x^{3}+y^{3}, -3y^{3}+2z^{4}, -6y^{2}x+y^{4}, -6yx^{2}+y^{3}x\}$ .
,$I_{1}=\langle x +y+6, z^{2}+18, y^{2}+6y+36\rangle$ , $\sqrt{I_{1}}=\langle z^{2}+18, x\mathrm{f}y+6, y^{2}+6y+36\rangle$ ,$I_{2}=\langle y-6, x -6, z^{2}+18\rangle$ , $\sqrt{I_{2}}=\langle z^{2}+18, y-6, x -6\rangle$ ,$I_{\mathit{3}}=\langle 3x^{2}+z^{2}y, z^{2}x+3y^{2}, zy^{2}, zyx +2z^{3}, zx^{2}, y^{2}x, yx^{2}, x^{3}-y^{3},3y^{3}-2z^{4}, y^{4}\rangle$ ,,$I=I_{1}\cap I_{2}\cap I_{3}$$\sqrt{I_{3}}=\langle z, y, x\rangle$.

$Ann^{(1)}=\langle f_{l}, f_{y}, f_{z}, P_{1}, P_{2}, P_{3}\rangle$$P_{1}=$$(48x^{2}-11y^{2}x-6z^{2}y)\partial_{l}+(60yx -10y^{3})\partial_{y}+(36zx -6zy^{2})\partial_{z}+324\mathrm{x}$ $-69y^{2}$$P_{2}=$$(4x^{3}-24yx)\partial_{ae}+(5yx^{2}+6z^{2}x-12y^{2})\partial_{y}+(3zx^{2}-18zy)\partial_{z}+30x^{2}-126y$$P\epsilon$$=$$(26zyx^{2}-576zx -60zy^{2})\partial_{l}+(29zy^{2}x+(18z^{3}-720z)y)\partial_{y}+((16z^{2}+72)yx -432z^{2})\partial_{z}$$-362z^{3}-3888z$$Ann^{(2)}=\langle f_{l}, f_{y}, f_{z}, P\rangle$

$P=$

,

,

.

,

$(864x -144y^{2})\partial_{l}\partial_{y}+(-2x^{2}-276y)\partial_{l}+(-144x^{2}+864y)\partial_{y}^{2}+(24z^{3}+432z)\partial_{y}\partial_{z}$$+(yx+218z^{2}+5616)\partial_{y}+(-36x+6y^{2})\partial_{z}^{2}-3x$.
.

$\sigma=[ -\frac{1}{2^{5}3^{6}}\frac{1}{xyz}+\frac{1}{2^{4}3^{4}}\frac{1}{xyz^{3}}-\frac{1}{2^{3}3^{2}}\frac{1}{xyz^{5}}-\frac{1}{2^{2}3^{3}}\frac{1}{xy^{4}z}-\frac{1}{2^{3}3^{4}}\frac{1}{x^{2}y^{2}z}+\frac{1}{2^{2}3^{2}}\frac{1}{x^{2}y^{2}z^{3}}-\frac{1}{2^{2}3^{3}}\frac{1}{x^{4}yz}]$

5Bimodal singularity2$E_{18}$に関する計算$\ovalbox{\tt\small REJECT}\ovalbox{\tt\small REJECT}*\mathrm{f}\mathrm{f}\mathrm{i}\text{斉}\Re \text{多}\mathrm{F}\text{式}f(x, y)=x^{3}+y^{10}+xy^{7}+xy^{8}\sigma)\mathrm{f}\mathrm{f}d)6$$\frac{\backslash }{*}.b$,$\mathrm{f}4\text{と}\overline{|\mathrm{p}\neg}\mathrm{f}\mathrm{f}\mathrm{i}\text{の_{}\beta}^{\ni+\mathrm{f}1\epsilon\acute{1}\overline{\mathrm{T}}^{\vee}\supset f^{\vee}}.$$Ann^{(1)}\epsilon$

ffl$\iota$.
$[1/f_{l}f_{y}]|(0,0)$$\backslash (\overline{\mathrm{c}}\star \mathrm{f}\mathrm{f}\text{数的}\backslash \text{局}\overline{\mathrm{p}}ffi\text{コ_{}J\backslash }+\text{モロ}$ $\backslash j^{\backslash }\backslash -\mathrm{f}\mathrm{f}\mathrm{i}$$E_{18}\#\mathit{4}$k%Bimodal WIEW ,$\mathrm{b}$$\mathrm{J}\mathrm{Y}6\text{と}$,$\Re \text{の}\not\equiv \mathrm{f}\mathrm{f}\mathrm{l}\epsilon_{\{}^{J}\S 6$$Ann^{(1)}\text{を}$.

$[a \frac{1}{xy}+b\frac{1}{xy^{2}}+\frac{52326274982537898625543}{3^{10}10^{20}}\frac{1}{xy^{3}}+\frac{354285915106436807093}{3^{9}10^{18}}\frac{1}{xy^{4}}$272111019228806143 13787973405502307 1

13210502493257 1$+–\overline{3^{8}10^{16}}x1\overline{8\mathrm{f}7}273431438893120571\overline{3^{7}10^{14}}\overline{xy^{6}}\overline{3^{6}10^{12}}\overline{xy^{7}}$$+ \frac{14232708293}{3^{5}10^{10}}\frac{1}{xy^{8}}++-\overline{3^{4}10^{8}}\overline{xy^{9}}\overline{3^{3}10^{6}}\overline{xy^{10}}\overline{3^{2}10^{4}}\overline{xy^{11}}$7124652664985303778499 175715525087212649 1$-\overline{3}\cdot$$10^{2}\overline{xy^{12}}-\overline{3^{9}10^{17}}\overline{x^{2}y}\overline{3^{8}10^{15}}\overline{x^{2}y^{2}}-$107457356825701

11264904961551 11815069901

1$+++\overline{3^{7}10^{13}}\overline{x^{2}y^{3}}\overline{3^{6}10^{11}}\overline{x^{2}y^{4}}\overline{3^{5}10^{9}}\overline{x^{2}y^{5}}$11224249 145899 149

11170850911193 1$– \overline{3^{4}10^{7}}\overline{x^{2}y^{6}}\overline{3^{3}10^{5}}\overline{x^{2}y^{7}}\overline{30}\overline{x_{7}^{2}y^{9}}-\frac{49}{3^{2}10^{3}}\frac{1}{x^{2}y^{8}}+-\overline{3^{6}10^{10}}\overline{x^{3}y}$320395243 1

178207 1

4157

7

1

1

251

1

$-+++ \frac{1}{x^{3}y^{5}}--\frac{1}{3^{2}10}\frac{1}{x^{4}y^{2}}\overline{3^{5}10^{8}}\overline{x^{3}y^{2}}\overline{3^{4}10^{6}}\overline{x^{3}y^{8}}\overline{3310^{4}}\overline{x^{3}y^{4}}\overline{3^{2}10^{2}}\overline{3^{3}10^{3}}\overline{x^{4}y}]$, 1$E_{18}\text{の}\ovalbox{\tt\small REJECT}_{\square }^{\mathrm{A}}$$\sim-\text{と}$$\emptyset*\mathrm{l}\text{分}\hslash\backslash 6$

.

\yen

ffffl

$\mathrm{r}\not\in \text{の微分}\dagger \mathrm{f}\mathrm{i}\mathrm{f}\mathrm{f}\mathrm{f}\mathrm{f}\mathrm{l}\ovalbox{\tt\small REJECT} k$$\text{と}b$$\ovalbox{\tt\small REJECT}!\ovalbox{\tt\small REJECT}\ovalbox{\tt\small REJECT}$$\text{と}$,)$*\mathcal{O}$$(\mathrm{i})$$\mathrm{V}$$\backslash f_{\beta}-.3+\mathrm{E}T|\Sigma[1/axy]\text{と}$, (\"u)$\mathrm{g}’\# 6$$[1/xy^{2}]\mathrm{C}1)\ !8$.

(i) ホロノミック系 $D_{X}/Ann^{(1)}$ の原点における重複度 $=3$(\"u)

$HoM_{\mathrm{X}}$ $(D_{X}/Ann^{(1)},\mathcal{H}_{[(0,0)]}^{2}(O\mathrm{x}))=\mathrm{S}\mathrm{p}\mathrm{a}\mathrm{n}\{[1/xy], [1/xy^{2}], \sigma\}$$a$, $bB\Re b$.

$6_{\sim}-\text{と}$$[] \mathrm{g}\tau\doteqdot rx\mathrm{t}$$\backslash$165
このことは, 特異点の 複雑さ とホロノミック系とを示唆していると思われる.
$\ovalbox{\tt\small REJECT}$$\ovalbox{\tt\small REJECT}$$\mathcal{D}x/Ann^{(\mathfrak{h}}$の重複度の間には何らかの関係があるこ

References[1] V.I. Arnol’d, Critical points of smooth functions and their normal forms, Russian Math. Surveys 30,
5 (1975), 1-75.

[2] Y. Nakamura, Construction of a system of differential operators as annihilators of a cohomology class-in connection with quasihomogeneous singularities-, Josai Mathematical Monographs 2(2000), 139-

148.

[3] 中村弥生, 田島慎一, 代数的局所コホモロジー類の満たすホロノミック系の構成法について, 京都大学
数理解析研究所講究録 「数式処理における理論と応用の研究」 , 掲載予定.
[4] 田島慎一, 中村弥生, 多変数有理関数の留数計算について, 京都大学数理解析研究所講究録「数式処理における理論と応用の研究」 , 1085(1999), 71-81.
[5] 田島慎一, 中村弥生, 擬斉次孤立特異点の標準形に対する双対基底の計算, 京都大学数理解析研究所講究録 「 -加群のアルゴリズム」, 1171(2000), 164-189.
$\mathrm{D}$[6] N.

$-\backslash \mathrm{T}$akayama, Kan: A system for computation in algebraic analysis (1991-), (. openxm$\mathrm{h}\mathrm{t}\mathrm{t}\mathrm{p}://\mathrm{w}\mathrm{w}\mathrm{w}$