|
gap> F:=GF(5);
GF(5)
gap> r:=PolynomialRing(F,4);
GF(5)[x_1,x_2,x_3,x_4]
gap> x:=IndeterminatesOfPolynomialRing(r);
[ x_1, x_2, x_3, x_4 ]
gap> pg:=PG(3,F);
ProjectiveSpace(3, 5)
gap> Q:=x[2]*x[3]+x[4]^2;
x_2*x_3+x_4^2
gap> qv:=QuadraticVariety(pg,Q);
Quadratic Variety in ProjectiveSpace(3, 5)
gap> AsSet(List(Planes(pg),z->Size(Filtered(Points(z),x->x in qv))));
[ 1, 6, 11 ]
gap> qf:=QuadraticForm(qv);
< quadratic form >
gap> Display(qf);
Quadratic form
Gram Matrix:
. . . .
. . 1 .
. . . .
. . . 1
Polynomial: [ [ x_2*x_3+x_4^2 ] ]
gap> IsDegenerateForm(qf);
#I Testing degeneracy of the *associated bilinear form*
true
gap> qv:=QuadraticVariety(3,F,"-");
Quadratic Variety in ProjectiveSpace(3, 5)
gap> PolarSpace(qv);
<polar space in ProjectiveSpace(3,GF(5)): x_1^2+Z(5)*x_2^2+x_3*x_4=0 >
gap> Display(last);
<polar space of rank 3 over GF(5)>
Non-singular elliptic quadratic form
Gram Matrix:
1 . . .
. 2 . .
. . . 1
. . . .
Polynomial: [ [ x_1^2+Z(5)*x_2^2+x_3*x_4 ] ]
Witt Index: 1
Bilinear form
Gram Matrix:
2 . . .
. 4 . .
. . . 1
. . 1 .
gap> qv:=QuadraticVariety(3,F,"+");
Quadratic Variety in ProjectiveSpace(3, 5)
gap> Display(last);
Quadratic Variety in ProjectiveSpace(3, 5)
Polynomial: [ Z(5)*x_1*x_2+Z(5)*x_3*x_4 ]
[ Dauer der Verarbeitung: 0.13 Sekunden
(vorverarbeitet)
]
|
