数理解析研究所講究録
第 1815 巻 2012 年 41-45

41

非特異 4 次曲線の判別式について
甲南大学・知能情報学部 高橋 正 (Tadashi Takahashi)
Department of Intelligence and Informatics, Konan University
概要

消去イデアルは，幅広い応用がある。 イデアルに関する様々な操作では，’ 解を得ることが$\prime$

できる

1

$=$

消去イデアルの計算ができる’/

が鍵になっている。 そのような例のーつを示す。

目的$\mathbb{P}^{2}$$f=x^{2}z+y^{3}+pyz^{2}+qz^{3}$ (p, q: パラメータ) とすると，における 3 次曲線については，$(f, \frac{\partial f}{\partial x}, \frac{\partial f}{\partial y}, \frac{\partial f}{\partial z})=(0,0,0,0)$より，曲線が非特異であるためには，$4p^{3}+27q^{2}\neq 0$という条件が得られる。 この現象をグレブナー基底を用いて示すと，以下のようになる。
$G(f, fi, f_{2}, f_{3})$を $f,$ $fi,$$f_{2}$, たのグレブナー基底とし，$f=x^{2}z+y^{3}+pyz^{2}+qz^{3},$から $x,$ $y,$$fi= \frac{\partial f}{\partial x},$$f_{2}= \frac{\partial f}{\partial y},$$f_{3}= \frac{\partial f}{\partial z}$とすると，を消去した消去イデアルは，$\{4p^{3}+27q^{2}\}$ となる。
そのため， における 3 次曲線が非特異になるためには，$4p^{3}+27q^{2}\neq 0$ という条件が得られる。 この 3 次曲線の現象と同様に，非特異 4 次曲線について，その定義方程式のパラメータが満た$G(f, fi, f_{2}, f_{3})$$z$$\mathbb{P}^{2}$

す条件式が判別式である。 判別式を消去イデアルを用いた計算によって求める。

非特異 4 次曲線の定義方程式2$\mathbb{P}^{2}$における非特異 4 次曲線の定義方程式としては，$x^{3}z+y^{3}z+pxyz^{2}+qxz^{3}+rxy^{4}+sy^{3}z+ty^{2}z^{2}+uyz^{3}+vz^{4}=0$$(p, q, r, s,t, u, v : パラメータ )$ (Case 1)と$x^{3}z+pxy^{2}z+pxyz^{2}+rxz^{3}+y^{4}+sy^{2}z^{2}+tyz^{3}+uz^{4}=0$$(p, q,r, s,t,u: パラメータ )$ (Case 2)の 2 種類がある。 本稿では，Case 2 の判別式を導出する。

3

消去イデアルの応用$G(f,f_{1},f_{2},f_{3})$を，$f,f_{1},f_{2},f_{3}$から構成されるグレブナー基底， を上記 Case2 の定義方程式，$f$$f1= \frac{\partial f}{\partial x}, f_{2}=\frac{\partial f}{\partial y}, f3=\frac{\partial f}{\partial z}$とするとき，Case 2 の非特異 4 次曲線の判別式を以下の手順よって求める。

42

.
.
.
.

$z$

を消去した消去イデアルを計算する$G(f, fi, f_{2}, f_{3})$ から $x,$ $z$を消去した消去イデアルを計算する$G(f, fi, f_{2}, f_{3})$ から $x,$ $y$を消去した消去イデアルを計算する$G(f, fi, f_{2}, f_{3})$から $y,$

上記 3 つの消去イデアルにおける共通解を見いだす

4

計算$var1$ , var2, . . . , varn を消去した消去イデアルの$G(f, fi, f_{2}, f_{3})$ から，Mathematica を用いて，計算を，以下のように行う。

$f=x^{3}z+pxy^{2}z+qxyz^{2}+rxz^{3}+y^{4}+sy^{2}z^{2}+tyz^{3}+uz^{4}$ ;Factor[GroebnerBasis
EliminationOrder]
MonomialOrder
Factor[GroebnerBasis
EliminationOrder]
MonomialOrder
Factor[GroebnerBasisMonomialOrder $arrow EliminationOrder$ ]$[\{f, \partial_{x}f, \partial_{y}f, \partial_{z}f\},$$arrow$$[\{f, \partial_{x}f, \partial_{y}f, \partial_{z}f\},$$arrow$$\{x,p, q, r, s, t, u\},$ $\{y, z\},$$]$$\{y,p, q, r, s, t, u\},$ $\{x, z\},$$]$$[\{f, \partial_{x}f, \partial_{y}f, \partial_{z}f\},$$\{z,p, q, r, s, t, u\},$ $\{x, y\},$$]$

その結果は，$G(f, f_{1}, f_{2}, f_{3})\cap I[x,p, q, r, s, t, u]=Dx^{7}$$G(f, f_{1}, f_{2}, f_{3})\cap I[y,p, q, r, s, t, u]=Dy^{7}$$G(f, f_{1}, f_{2}, f_{3})\cap I[z,p, q, r, s, t, u]=Dz^{7}$

ここで、 $D$ は，$p,$ $q,$ $r,$ $s,$$t,$ $u$の多項式となることを得る。
$D$ であることが分かる。
したがって，Case2 の定義方程式の判別式は，$D$は，以下のようになる。

$D=-6912u^{4}p^{12}-1024r^{3}u^{2}p^{12}+13824rsu^{3}p^{11}+13824qtu^{3}p^{11}-1024r^{4}t^{2}p^{11}+768q^{2}r^{2}u^{2}p^{11}-$$9216rt^{2}u^{2}p^{11}+2048r^{4} \sup^{11}+1024qr^{3}tup^{11}-2048r^{2}t^{4}p^{10}-13824r^{2}u^{3}p^{10}-10368q^{2}su^{3}p^{10}-$$1024r^{5}s^{2}p^{10}+768q^{2}r^{3}t^{2}p^{10}-6912r^{2}s^{2}u^{2}p^{10}-8064q^{2}t^{2}u^{2}p^{10}-192q^{4}ru^{2}p^{10}-2304qrstu^{2}p^{10}+$$1024qr^{4}stp^{10}-2048r^{5}up^{10}+9216qrt^{3}up^{10}+3072r^{2}st^{2}up^{10}-2560q^{2}r^{3} \sup^{10}-768q^{3}r^{2}tup^{10}-$$1024t^{6}p^{9}-1280q^{2}rt^{4}p^{9}-497664su^{4}p^{9}+6656qr^{2}st^{3}p^{9}+13824s^{3}u^{3}p^{9}+20736t^{2}u^{3}p^{9}+27648q^{2}ru^{3}p^{9}+$$768q^{2}r^{4}s^{2}p^{9}-192q^{4}r^{2}t^{2}p^{9}-3840r^{3}s^{2}t^{2}p^{9}+16q^{6}u^{2}p^{9}+4608q^{2}rs^{2}u^{2}p^{9}-24192s^{2}t^{2}u^{2}p^{9}-59904r^{3}su^{2}p^{9}-$$6912qr^{2}tu^{2}p^{9}+10944q^{3}stu^{2}p^{9}+2048r^{6}sp^{9}-3072qr^{5}tp^{9}-768q^{3}r^{3}stp^{9}+5632q^{2}r^{4}up^{9}+9216st^{4}up^{9}+$$2048r^{3}s^{3}up^{9}+1152q^{3}t^{3} up^{9}-1536r^{3}t^{2}up^{9}-14208q^{2}rst^{2}up^{9}+1152q^{4}r^{2}\sup^{9}+768qr^{2}s^{2}tup^{9}+$$192q^{5}$

rtup9 $-1024r^{7}p^{8}+1536qst^{5}p^{8}-2048r^{4}s^{4}p^{8}+16q^{4}t^{4}p^{8}-768rs^{2}t^{4}p^{8}+995328ru^{4}p^{8}-$$3328qr^{3}t^{3}p^{8}+1856q^{3}rst^{3}p^{8}-7776q^{4}u^{3}p^{8}+912384rs^{2}u^{3}p^{8}+829440qstu^{3}p^{8}-192q^{4}r^{3}s^{2}p^{8}-$$7296q^{2}r^{2}s^{2}t^{2}p^{8}+16q^{6}rt^{2}p^{8}-61440r^{4}st^{2}p^{8}+140544r^{4}u^{2}p^{8}-13824rs^{4}u^{2}p^{8}-3456qt^{3}u^{2}p^{8}-$$3312q^{4}s^{2}u^{2}p^{8}-590976rst^{2}u^{2}p^{8}+41472q^{2}r^{2}su^{2}p^{8}+27648qs^{3}tu^{2}p^{8}-18432q^{3}rtu^{2}p^{8}-2560q^{2}r^{5}sp^{8}+$$3328q^{3}r^{4}tp^{8}+6656qr^{3}s^{3}tp^{8}+192q^{5}r^{2}stp^{8}+4608rt^{4}up^{8}-4480q^{4}r^{3}up^{8}-1920q^{2}r^{2}s^{3}up^{8}-12672qs^{2}t^{3}up^{8}+$$135168r^{4}s^{2}up^{8}+5760rs^{3}t^{2}up^{8}+23040q^{2}r^{2}t^{2}up^{8}-1824q^{4}st^{2}up^{8}-224q^{6}$

rsup8 $-16q^{7}tup^{8}+$$7680q^{3}rs^{2}tup^{8}+44544qr^{3}stup^{8}+768q^{2}r^{6}p^{7}-2304qrt^{5}p^{7}-1280q^{2}r^{3}s^{4}p^{7}-480q^{2}s^{2}t^{4}p^{7}-123648r^{2}st^{4}p^{7}-$43$1741824q^{2}u^{4}p^{7}-59392r^{5}s^{3}p^{7}+192qrs^{3}t^{3}p^{7}-896q^{3}r^{2}t^{3}p^{7}-16q^{5}st^{3}p^{7}-518400q^{2}s^{2}u^{3}p^{7}-2820096r^{2}su^{3}p^{7}-$$1741824qrtu^{3}p^{7}+16q^{6}r^{2}s^{2}p^{7}+135936r^{5}t^{2}p^{7}-768r^{2}s^{4}t^{2}p^{7}-52Sq^{4}rs^{2}t^{2}p^{7}+5452Sq^{2}r^{3}st^{2}p^{7}-$$10368q^{2}s^{4}u^{2}p^{7}-368640q^{2}r^{3}u^{2}p^{7}-400896r^{2}s^{3}u^{2}p^{7}+1264S96r^{2}t^{2}u^{2}p^{7}-4S9SSSq^{2}st^{2}u^{2}p^{7}-576q^{4}rsu^{2}p^{7}+$$7632q^{5}tu^{2}p^{7}-228096qrs^{2}tu^{2}p^{7}+1152q^{4}r^{4}sp^{7}-1344q^{5}r^{3}tp^{7}+1S56q^{3}r^{2}s^{3}tp^{7}+49152qr^{4}s^{2}tp^{7}-$$16q^{7}rstp^{7}+576q^{2}t^{4}up^{7}-1440q^{4}rs^{3}up^{7}+572544qrst^{3}up^{7}+1568q^{6}r^{2}up^{7}-139008q^{2}r^{3}s^{2}up^{7}+$$3744q^{2}s^{3}t^{2}up^{7}+1 S6624r^{2}s^{2}t^{2}up^{7}-604Sq^{4}rt^{2}up^{7}+16q^{8}\sup^{7}-417792r^{5}\sup^{7}-103680qr^{4}tup^{7}+$$2304qrs^{4}tup^{7}+624q^{5}s^{2}tup^{7}-56064q^{3}r^{2}stup^{7}-1024r^{3}s^{6}p^{6}-62208st^{6}p^{6}-192q^{4}r^{5}p^{6}+1056q^{3}t^{5}p^{6}+$$5971968u^{5}p^{6}+16q^{4}r^{2}s^{4}p^{6}-432s^{4}t^{4}p^{6}+277440r^{3}t^{4}p^{6}-77856q^{2}rst^{4}p^{6}-11943936s^{2}u^{4}p^{6}+40448q^{2}r^{4}s^{3}p^{6}-$$16q^{3}s^{3}t^{3}p^{6}+389760qr^{2}s^{2}t^{3}p^{6}+1360q^{5}rt^{3}p^{6}+870912s^{4}u^{3}p^{6}+2764S00r^{3}u^{3}p^{6}+1119744st^{2}u^{3}p^{6}+$$6013440q^{2}rsu^{3}p^{6}+32140S0q^{3}tu^{3}p^{6}+25S04Sr^{6}s^{2}p^{6}-32S320q^{2}r^{4}t^{2}p^{6}-4S0q^{2}rs^{4}t^{2}p^{6}-228S64r^{3}s^{3}t^{2}p^{6}-$$16608q^{4}r^{2}st^{2}p^{6}-6912s^{6}u^{2}p^{6}+1S144t^{4}u^{2}p^{6}+2522SSq^{2}rs^{3}u^{2}p^{6}+219456q^{4}r^{2}u^{2}p^{6}-2764Sr^{3}s^{2}u^{2}p^{6}-$$1482624s^{3}t^{2}u^{2}p^{6}-1145664q^{2}rt^{2}u^{2}p^{6}-2S0Sq^{6}su^{2}p^{6}+620352q^{3}s^{2}tu^{2}p^{6}-224q^{6}r^{3}sp^{6}+1536qr^{2}s^{5}tp^{6}-$$16q^{5}rs^{3}tp^{6}+240q^{7}r^{2}tp^{6}-36480q^{3}r^{3}s^{2}tp^{6}-314880qr^{5}stp^{6}+278528r^{6}up^{6}-1152q^{2}rs^{5}up^{6}+$$129024r^{3}s^{4}up^{6}+561600s^{2}t^{4}up^{6}+16q^{6}s^{3}up^{6}-1195776qr^{2}t^{3}up^{6}+61344q^{3}st^{3}up^{6}+61248q^{4}r^{2}s^{2}up^{6}-$$1200q^{6}t^{2}up^{6}+3456s^{5}t^{2}up^{6}-791424q^{2}rs^{2}t^{2}up^{6}-542592r^{3}st^{2}up^{6}-256q^{8}rup^{6}+1127040q^{2}r^{4} \sup^{6}+$

5 $76q^{3}s^{4}tup^{6}+305664q^{3}r^{3}tup^{6}+2764Sqr^{2}s^{3}tup^{6}+17664q^{5}rstup^{6}+13996Srt^{6}p^{5}-116736r^{4}s^{5}p^{5}+$$85536qs^{2}t^{5}p^{5}+16q^{6}r^{4}p^{5}-44064rs^{3}t^{4}p^{5}-277776q^{2}r^{2}t^{4}p^{5}+456q^{4}st^{4}p^{5}+29S59S40rsu^{4}p^{5}-$$41803776qtu^{4}p^{5}-116S0q^{4}r^{3}s^{3}p^{5}-16q^{7}t^{3}p^{5}+103296q^{3}rs^{2}t^{3}p^{5}-1091904qr^{3}st^{3}p^{5}+16920576rs^{3}u^{3}p^{5}-$$7050240q^{2}r^{2}u^{3}p^{5}+5474304rt^{2}u^{3}p^{5}-25S1632q^{4}su^{3}p^{5}+14929920qs^{2}tu^{3}p^{5}-463104q^{2}r^{5}s^{2}p^{5}+$$196800q^{4}r^{3}t^{2}p^{5}-407232q^{2}r^{2}s^{3}t^{2}p^{5}-536832r^{4}s^{2}t^{2}p^{5}+40Sq^{6}rst^{2}p^{5}-7S796Srs^{5}u^{2}p^{5}-190944q^{4}s^{3}u^{2}p^{5}-$$414720qst^{3}u^{2}p^{5}-1555200q^{2}r^{2}s^{2}u^{2}p^{5}-1S13104q^{4}t^{2}u^{2}p^{5}-S335S72rs^{2}t^{2}u^{2}p^{5}-47S0Sq^{6}ru^{2}p^{5}+$$2211S40r^{4}su^{2}p^{5}+1534464qs^{4}tu^{2}p^{5}-4561920qr^{3}tu^{2}p^{5}-29S59S4q^{3}rstu^{2}p^{5}-3276S0r^{7}sp^{5}+16q^{8}r^{2}sp^{5}+$$389120qr^{6}tp^{5}+374784qr^{3}s^{4}tp^{5}+12192q^{5}r^{2}s^{2}tp^{5}-16q^{9}rtp^{5}+493632q^{3}r^{4}stp^{5}+16q^{10}up^{5}$-$1155072q^{2}r^{5}up^{5}+23328qt^{5}up^{5}-110592q^{2}r^{2}s^{4}up^{5}-1003104rst^{4}up^{5}+2506752r^{4}s^{3}up^{5}-705024qs^{3}t^{3}up^{5}+$$1910592q^{3}rt^{3}up^{5}-13584q^{6}rs^{2}up^{5}+1142784r^{4}t^{2}up^{5}+331776rs^{4}t^{2}up^{5}-9S064q^{4}s^{2}t^{2}up^{5}+3755808q^{2}r^{2}st^{2}up^{5}-$$938304q^{4}r^{3} \sup^{5}+428544q^{3}rs^{3}tup^{5}-194112q^{5}r^{2}tup^{5}+117504qr^{3}s^{2}tup^{5}-312q^{7}stup^{5}+131072r^{8}p^{4}-$$227448q^{2}t^{6}p^{4}-70656q^{2}r^{3}s^{5}p^{4}-33S256qrst^{5}p^{4}-669696r^{5}s^{4}p^{4}-2656Sq^{2}s^{3}t^{4}p^{4}-2056752r^{2}s^{2}t^{4}p^{4}-$$287100q^{4}rt^{4}p^{4}-29S59S40r^{2}u^{4}p^{4}-22394SS0q^{2}su^{4}p^{4}+SSSq^{6}r^{2}s^{3}p^{4}-3S6S416qr^{4}t^{3}p^{4}+1036Sqrs^{4}t^{3}p^{4}-$$888q^{5}s^{2}t^{3}p^{4}+1132464q^{3}r^{2}st^{3}p^{4}-2737152q^{2}s^{3}u^{3}p^{4}-80621568r^{2}s^{2}u^{3}p^{4}+6954S544q^{2}t^{2}u^{3}p^{4}+$$6863616q^{4}ru^{3}p^{4}+99532S0qrstu^{3}p^{4}+245136q^{4}r^{4}s^{2}p^{4}-41472r^{2}s^{5}t^{2}p^{4}-29232q^{4}rs^{3}t^{2}p^{4}-45552q^{6}r^{2}t^{2}p^{4}+$$1440288q^{2}r^{3}s^{2}t^{2}p^{4}+3950592r^{5}st^{2}p^{4}+32S5q^{8}u^{2}p^{4}-353S944r^{5}u^{2}p^{4}-570240q^{2}s^{5}u^{2}p^{4}-452044Sr^{2}s^{4}u^{2}p^{4}-$46407168 $qrt^{3}u^{2}p^{4}+1767744q^{4}rs^{2}u^{2}p^{4}-139034SSq^{2}s^{2}t^{2}u^{2}p^{4}+40103424r^{2}st^{2}u^{2}p^{4}-2419200q^{2}r^{3}su^{2}p^{4}-$$10948608qrs^{3}$ tu $2_{p}4+48314SSq^{3}r^{2}tu^{2}p^{4}+270604Sq^{5}stu^{2}p^{4}+796672q^{2}r^{6}sp^{4}-1060608q^{3}r^{5}tp^{4}+$$102528q^{3}r^{2}s^{4}tp^{4}-511488qr^{4}s^{3}tp^{4}-888q^{7}rs^{2}tp^{4}-251616q^{5}r^{3}stp^{4}+1734S16q^{4}r^{4}up^{4}-794SSq^{4}rs^{4}up^{4}+$$1236384r^{2}t^{4}up^{4}+2058696q^{2}st^{4}up^{4}-123S400q^{2}r^{3}s^{3}up^{4}+255SSSq^{5}t^{3}up^{4}+12389760qrs^{2}t^{3}up^{4}+$$888q^{8}s^{2}up^{4}-11943936r^{5}s^{2}up^{4}+207360q^{2}s^{4}t^{2}up^{4}+365760q^{2}r^{3}t^{2}up^{4}+2498688r^{2}s^{3}t^{2}up^{4}-3066552q^{4}rst^{2}up^{4}+$$350448q^{6}r^{2} \sup^{4}+124416qrs^{5}tup^{4}+34560q^{5}s^{3}tup^{4}-2223936q^{3}r^{2}s^{2}tup^{4}+45792q^{7}rtup^{4}+744652Sqr^{4}stup^{4}-$311 $296q^{2}r^{7}p^{3}-55296r^{3}s^{7}p^{3}-1102248s^{2}t^{6}p^{3}+S64q^{4}r^{2}s^{5}p^{3}-S223120qr^{2}t^{5}p^{3}+37S10Sq^{3}st^{5}p^{3}+$214990848 $su^{5}p^{3}+272640q^{2}r^{4}s^{4}p^{3}+3525q^{6}t^{4}p^{3}-2332Ss^{5}t^{4}p^{3}-1473552q^{2}rs^{2}t^{4}p^{3}+S273664r^{3}st^{4}p^{3}-$$113467392s^{3}u^{4}p^{3}-116453376t^{2}u^{4}p^{3}+597196Sq^{2}ru^{4}p^{3}+576716Sr^{6}s^{3}p^{3}-S64q^{3}s^{4}t^{3}p^{3}+264220Sq^{3}r^{3}t^{3}p^{3}+$$6397056qr^{2}s^{3}t^{3}p^{3}+462S2Sq^{5}rst^{3}p^{3}-1546560q^{6}u^{3}p^{3}+16422912s^{5}u^{3}p^{3}+7564492Sq^{2}rs^{2}u^{3}p^{3}+$$37324800s^{2}t^{2}u^{3}p^{3}+1592524SOr^{3}su^{3}p^{3}+29S59S4qr^{2}tu^{3}p^{3}-35334144q^{3}stu^{3}p^{3}-53520q^{6}r^{3}s^{2}p^{3}-$$4767744r^{6}t^{2}p^{3}-25920q^{2}rs^{5}t^{2}p^{3}-3794688r^{3}s^{4}t^{2}p^{3}-1915416q^{4}r^{2}s^{2}t^{2}p^{3}+3525q^{8}rt^{2}p^{3}-606720q^{2}r^{4}st^{2}p^{3}-$44$373248s^{7}u^{2}p^{3}+2764800q^{2}r^{4}u^{2}p^{3}+2198016q^{2}rs^{4}u^{2}p^{3}-7558272st^{4}u^{2}p^{3}+22118400r^{3}s^{3}u^{2}p^{3}-$$35323776q^{3}t^{3}u^{2}p^{3}-860976q^{6}s^{2}u^{2}p^{3}-26687232s^{4}t^{2}u^{2}p^{3}-61378560r^{3}t^{2}u^{2}p^{3}+12234240q^{2}rst^{2}u^{2}p^{3}-$$943488q^{4}r^{2}su^{2}p^{3}+15344640q^{3}s^{3}tu^{2}p^{3}-10782720qr^{2}s^{2}tu^{2}p^{3}-5759424q^{5}rtu^{2}p^{3}-681216q^{4}r^{5}sp^{3}+$82944 $qr^{2}s^{6}tp^{3}+S4S496q^{5}r^{4}tp^{3}-864q^{5}rs^{4}tp^{3}+816000q^{3}r^{3}s^{3}tp^{3}-9560064qr^{5}s^{2}tp^{3}+54120q^{7}r^{2}stp^{3}-$$62208q^{2}rs^{6}up^{3}+944784t^{6}up^{3}+2433024r^{3}s^{5}up^{3}+864q^{6}s^{4}up^{3}+10007712s^{3}t^{4}up^{3}+40427424q^{2}rt^{4}up^{3}-$$1108096q^{6}r^{3}up^{3}+605664q^{4}r^{2}s^{3}up^{3}-1872288q^{3}s^{2}t^{3}up^{3}-30233088qr^{2}st^{3}up^{3}+17178624q^{2}r^{4}s^{2}up^{3}+$$186624s^{6}t^{2}up^{3}-11000448q^{2}rs^{3}t^{2}up^{3}+2467152q^{4}r^{2}t^{2}up^{3}-12386304r^{3}s^{2}t^{2}up^{3}-446958q^{6}st^{2}up^{3}+$$18874368r^{6} \sup^{3}-61770q^{8}r\sup^{3}-3525q^{9}tup^{3}-4866048qr^{5}tup^{3}+31104q^{3}s^{5}tup^{3}-476928qr^{2}s^{4}tup^{3}+$stup3-425l528qt 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