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gap> gf := GF(8);
GF(2^3)
gap> vec := gf^3;
( GF(2^3)^3 )
gap> r := PolynomialRing( gf, 3);
GF(2^3)[x_1,x_2,x_3]
gap> poly := r.1^2 + r.2 * r.3;
x_1^2+x_2*x_3
gap> form := QuadraticFormByPolynomial( poly, r );
< quadratic form >
gap> Display( form );
Quadratic form
Gram Matrix:
1 . .
. . 1
. . .
Polynomial: x_1^2+x_2*x_3
gap> IsDegenerateForm( form );
#I Testing degeneracy of the *associated bilinear form*
true
gap> IsSingularForm( form );
false
gap> WittIndex( form );
1
gap> IsParabolicForm( form );
true
gap> RadicalOfForm( form );
<vector space of dimension 0 over GF(2^3)>
gap> canonical := IsometricCanonicalForm( form );
< parabolic quadratic form >
gap> form = canonical;
true
gap> go := GO(3,8);
GO(0,3,8)
gap> mat := InvariantQuadraticForm( go )!.matrix;
[ [ Z(2)^0, 0*Z(2), 0*Z(2) ], [ 0*Z(2), 0*Z(2), 0*Z(2) ],
[ 0*Z(2), Z(2)^0, 0*Z(2) ] ]
gap> gapform := QuadraticFormByMatrix( mat, GF(8) );
< quadratic form >
gap> b := BaseChangeToCanonical( gapform );
[ [ Z(2)^0, 0*Z(2), 0*Z(2) ], [ 0*Z(2), Z(2)^0, 0*Z(2) ],
[ 0*Z(2), 0*Z(2), Z(2)^0 ] ]
gap> hom := BaseChangeHomomorphism( b, GF(8) );
^[ [ Z(2)^0, 0*Z(2), 0*Z(2) ], [ 0*Z(2), Z(2)^0, 0*Z(2) ],
[ 0*Z(2), 0*Z(2), Z(2)^0 ] ]
gap> newgo := Image(hom, go);
Group(
[
[ [ Z(2)^0, 0*Z(2), 0*Z(2) ], [ 0*Z(2), Z(2^3), 0*Z(2) ],
[ 0*Z(2), 0*Z(2), Z(2^3)^6 ] ],
[ [ Z(2)^0, 0*Z(2), 0*Z(2) ], [ Z(2)^0, Z(2)^0, Z(2)^0 ],
[ 0*Z(2), Z(2)^0, 0*Z(2) ] ] ])
gap> conic := Filtered(vec, x -> IsZero( x^form ));;
gap> Size(conic);
64
gap> orbs := Orbits(newgo, conic, OnRight);;
gap> List(orbs,Size);
[ 1, 63 ]
[ Dauer der Verarbeitung: 0.4 Sekunden
(vorverarbeitet)
]
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