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gap> #isotropic subspaces. quadratic forms
gap> q := 9;
9
gap> f := GF(q);
GF(3^2)
gap> dim := 3;
3
gap> v := f^dim;
( GF(3^2)^3 )
gap> mat := IdentityMat(dim,f);
[ [ Z(3)^0, 0*Z(3), 0*Z(3) ], [ 0*Z(3), Z(3)^0, 0*Z(3) ],
[ 0*Z(3), 0*Z(3), Z(3)^0 ] ]
gap> form := QuadraticFormByMatrix(mat,f);
< quadratic form >
gap> lines := Subspaces(v,1);
Subspaces( ( GF(3^2)^3 ), 1 )
gap> matrices := List(lines,x->BasisVectors(Basis(x)));;
gap> vectors := List(matrices,x->x[1]);;
gap> results := Collected(List(vectors,x->EvaluateForm(form,x)));;
gap> [Zero(f),(q^(dim-1)-1)/(q-1)] in results;
true
gap> results := Collected(List(matrices,x->x^form));;
gap> [[[Zero(f)]],(q^(dim-1)-1)/(q-1)] in results;
true
gap> Number(vectors,x->IsSingularVector(form,x))=(q^(dim-1)-1)/(q-1);
true
gap> Number(matrices,x->IsTotallySingularSubspace(form,x))=(q^(dim-1)-1)/(q-1);
true
gap> quit;
[ Dauer der Verarbeitung: 0.15 Sekunden
(vorverarbeitet)
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