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gap> f := GF(25);
GF(5^2)
gap> r := PolynomialRing(f,4);
GF(5^2)[x_1,x_2,x_3,x_4]
gap> ind := IndeterminatesOfPolynomialRing(r);
[ x_1, x_2, x_3, x_4 ]
gap> eq1 := Sum(List(ind,t->t^2));
x_1^2+x_2^2+x_3^2+x_4^2
gap> var := ProjectiveVariety(PG(3,f),[eq1]);
Projective Variety in ProjectiveSpace(3, 25)
gap> PolarSpace(var);
<polar space in ProjectiveSpace(3,GF(5^2)): x_1^2+x_2^2+x_3^2+x_4^2=0 >
gap> eq2 := Sum(List(ind,t->t^4));
x_1^4+x_2^4+x_3^4+x_4^4
gap> var := ProjectiveVariety(PG(3,f),[eq2]);
Projective Variety in ProjectiveSpace(3, 25)
gap> PolarSpace(var);
Error, <poly> does not generate a Hermitian matrix called from
GramMatrixByPolynomialForHermitianForm( pol, gf, n, vars ) called from
HermitianFormByPolynomial( pol, pring, n ) called from
HermitianFormByPolynomial( eq, r ) called from
<function "unknown">( <arguments> )
called from read-eval loop at line 16 of *stdin*
you can 'quit;' to quit to outer loop, or
you can 'return;' to continue
brk> quit;
gap> eq3 := Sum(List(ind,t->t^6));
x_1^6+x_2^6+x_3^6+x_4^6
gap> var := ProjectiveVariety(PG(3,f),[eq3]);
Projective Variety in ProjectiveSpace(3, 25)
gap> PolarSpace(var);
<polar space in ProjectiveSpace(3,GF(5^2)): x_1^6+x_2^6+x_3^6+x_4^6=0 >
[ Dauer der Verarbeitung: 0.17 Sekunden
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