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1.3 主な関数


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1.3.1 intersect

intersect(F,G)

:: 2曲線F=0,G=0 の交点の座標からなるリストを返す.

return

リスト

F G

変数x,y,z の斉次多項式

 
[1] intersect(y^2-x*z,(x^2+y^2)^3-4*x^2*y^2*z^2);
[[0,0,1],[(#4),(#5),1]]
[2] defpoly(alg(4));
t#4^3+3*t#4^2+3*t#4-3
[3] defpoly(alg(5));
t#5^2-t#4
[4] intersect(x^2-y^2,x^3+y*x^2+(y^2-z^2)*x+y^3-z^2*y);
***two curve have common components***

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1.3.2 sing

sing(F)

:: 曲線F=0 の特異点の座標からなるリストを返す.

return

リスト

F

変数x,y,z の斉次多項式

 
[1] sing(16*x^6-24*z^2*x^4+9*z^4*x^2+4*z^2*y^4-4*z^4*y^2);
[[0,0,1],[(#4),0,1],[1/2,(#3),1],[-1/2,(#3),1],[0,1,0]]
[2] defpoly(alg(3));
2*t#3^2-1
[3] defpoly(alg(4));
4*t#4^2-3
[4] sing((x-y)*(y^2-x*z));
[[1,1,1],[0,0,1]]
[5] sing((x-y)^2*(y^2-x*z));
***Argument has multiple divisor***
参照

nbh multia


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1.3.3 nbh

nbh(F)

:: 曲線F=0 のneighborhood graph を返す。

return

リスト

F

変数x,y,z の斉次多項式

参照

sing


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1.3.4 genus

genus(F)

:: 曲線F=0 の特異点の座標からなるリストを返す.

return

0以上の整数

F

変数x,y,z の斉次多項式

 
[1] genus(x^6+3*y^2*x^4+(3*y^4-4*z^2*y^2)*x^2+y^6);
0
[2] genus(y^2*z-x^3-z^3);
1
[3] genus(x^2+y^2+z^2-x*y-y*z-z*x);
-1
[4] fctr(x^2+y^2+z^2-x*y-y*z-z*x);
[[1,1],[x^2+(-y-z)*x+y^2-z*y+z^2,1]]
[5] irr_conic(x^2+y^2+z^2-x*y-y*z-z*x);
reducible
参照

irr_conic


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1.3.5 adjoint1,adjoint2

adjoint1(F)
adjoint2(F)

:: それぞれ曲線F=0のn-1次,n-2次の随伴曲線(adjoint curve)を返す(n=deg(F))。

return

線形のパラメーターを含む変数x,y,z の斉次多項式

F

変数x,y,z の斉次多項式

 
[1] adjoint2(x^6+3*y^2*x^4+(3*y^4-4*z^2*y^2)*x^2+y^6);
[c2,c3,c4,c6,c7] 5
(c2-c4)*x^4+c3*y*x^3+(c2*y^2+c6*z*y)*x^2+(c3*y^3+c7*z*y^2)*x+c4*y^4
[2] adjoint1(F);
[c1,c7,c11,c12,c13,c15,c16,c17,c18,c19,c20] 11
(c1*y+(c11-c15+c18-c20)*z)*x^4+(c13*y^2+c7*z*y+c11*z^2)*x^3+(c17*z*y^2+c12*z^2*y
+c15*z^3)*x^2+(c13*z^2*y^2+c16*z^3*y+c18*z^4)*x+c17*z^3*y^2+c19*z^4*y+c20*z^5
参照

restriction


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1.3.6 intpt

intpt(F)

:: 二次曲線F=0 上の整数点[x,y,z] をひとつ見つけて返す。整数点が存在しなければ、文字列no integer solutionを返す。

return

リスト、あるいは文字列no integer solution.

F

変数x,y,z の二次の斉次多項式

 
[1] intpt(22*x^2-10*y^2+z^2+5*x*y+13*y*x-z*x);
[71,-121,473]
[2] intpt(22*x^2-10*y^2+z^2+5*x*y+12*y*x-z*x);
no integer solution

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1.3.7 parametrize

parametrize(F)

:: 有理曲線F=0 をパラメトライズする多項式の組を返す。

return

リスト

F

有理曲線の定義多項式(変数x,y,z の斉次多項式)

 
[1] parametrize(x^4+(2*y^2-z^2)*x^2+y^4+z^2*y^2);
[-t^3-t,t^3-t,t^4+1,(-x^2-y^2)/(z*x+z*y)]
[2] parametrize((x^2+y^2)^3-4*x^2*y^2*z^2);
heuristic2 failed...
heuristic3 succeed
[32256*t^6-133120*t^5-129024*t^4+1064960*t^3-516096*t^2
-2129920*t+2064384,-127008*t^6+1048320*t^5-2671232*t^4
+10684928*t^2-16773120*t+8128512,274625*t^6-3194100*t^5
+15678780*t^4-41555808*t^3+62715120*t^2-51105600*t+17576000,
(-126*x^4+1040*y*x^3-382*y^2*x^2+1040*y^3*x-256*y^4)
/(-65*x^4+520*y*x^3+(-65*y^2-32*z*y)*x^2+(520*y^3+256*z*y^2)*x)]
[3] parametrize(22*x^2-10*y^2+z^2+5*x*y+12*y*x-z*x);
[(220*#6-10)*t^2+(-22*#6+1),(374*#6-17)*t^2+(-22*#6-43)*t,
(220*#6+210)*t^2+(-374*#6+17)*t+22,(-y)/((22*#6-1)*x+z)]
参照

genus


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