Copy examples to tests

tst/examples/assocbilform.tst

@@ -0,0 +1,20 @@
+gap> START_TEST("Forms: assocbilform.tst");
+gap> #Constructing form: AssociatedBilinearForm
+gap> r:= PolynomialRing(GF(121),6);
+GF(11^2)[x_1,x_2,x_3,x_4,x_5,x_6]
+gap> poly := r.1*r.5-r.2*r.6+r.3*r.4;
+x_1*x_5-x_2*x_6+x_3*x_4
+gap> form := QuadraticFormByPolynomial(poly,r);
+< quadratic form >
+gap> aform := AssociatedBilinearForm(form);
+< bilinear form >
+gap> Display(aform);
+Bilinear form
+Gram Matrix:
+  .  .  .  .  1  .
+  .  .  .  .  . 10
+  .  .  .  1  .  .
+  .  .  1  .  .  .
+  1  .  .  .  .  .
+  . 10  .  .  .  .
+gap> STOP_TEST("assocbilform.tst", 10000 );

tst/examples/basechangehom.tst

@@ -0,0 +1,28 @@
+gap> START_TEST("Forms: basechangehom.tst");
+gap> #morphisms: BaseChangeHomomorphism
+gap> gl:=GL(3,3);
+GL(3,3)
+gap> go:=GO(3,3);
+GO(0,3,3)
+gap> form := PreservedSesquilinearForms(go)[1]; 
+< bilinear form >
+gap> gram := GramMatrix( form );  
+[ [ 0*Z(3), Z(3)^0, 0*Z(3) ], [ Z(3)^0, 0*Z(3), 0*Z(3) ], 
+  [ 0*Z(3), 0*Z(3), Z(3) ] ]
+gap> b := BaseChangeToCanonical(form);;
+gap> hom := BaseChangeHomomorphism(b, GF(3));
+^[ [ 0*Z(3), 0*Z(3), Z(3) ], [ Z(3), Z(3), Z(3)^0 ], [ Z(3), 0*Z(3), Z(3) ] ]
+gap> newgo := Image(hom, go); 
+Group(
+[ 
+  [ [ Z(3)^0, 0*Z(3), Z(3) ], [ Z(3)^0, Z(3), Z(3)^0 ], 
+      [ 0*Z(3), 0*Z(3), Z(3) ] ], 
+  [ [ Z(3), Z(3)^0, 0*Z(3) ], [ 0*Z(3), Z(3), 0*Z(3) ], 
+      [ Z(3)^0, Z(3)^0, Z(3) ] ] ])
+gap> gens := GeneratorsOfGroup(newgo);;
+gap> canonical := b * gram * TransposedMat(b);
+[ [ Z(3), 0*Z(3), 0*Z(3) ], [ 0*Z(3), 0*Z(3), Z(3)^0 ], 
+  [ 0*Z(3), Z(3)^0, 0*Z(3) ] ]
+gap> ForAll(gens, y -> y * canonical * TransposedMat(y) = canonical);
+true
+gap> STOP_TEST("basechangehom.tst", 10000 );

tst/examples/basechangetocanonical.tst

@@ -0,0 +1,22 @@
+gap> START_TEST("Forms: basechangetocanonical.tst");
+gap> #morphisms: BaseChangeToCanonical
+gap> gf := GF(3);
+GF(3)
+gap> gram := [
+> [0,0,0,1,0,0], 
+> [0,0,0,0,1,0],
+> [0,0,0,0,0,1],
+> [-1,0,0,0,0,0],
+> [0,-1,0,0,0,0],
+> [0,0,-1,0,0,0]] * One(gf);;
+gap> form := BilinearFormByMatrix( gram, gf );
+< bilinear form >
+gap> b := BaseChangeToCanonical( form );;
+gap> Display( b * gram * TransposedMat(b) );
+ . 1 . . . .
+ 2 . . . . .
+ . . . 1 . .
+ . . 2 . . .
+ . . . . . 1
+ . . . . 2 .
+gap> STOP_TEST("basechangetocanonical.tst", 10000 );

tst/examples/bg_th_ex1.tst

@@ -0,0 +1,8 @@
+gap> START_TEST("Forms: bg_th_ex1.tst");
+gap> #Background theory: example 1
+gap> mat := [[1,0,0],[0,1,4],[1,2,1]]*Z(5)^0;
+[ [ Z(5)^0, 0*Z(5), 0*Z(5) ], [ 0*Z(5), Z(5)^0, Z(5)^2 ], 
+  [ Z(5)^0, Z(5), Z(5)^0 ] ]
+gap> form := BilinearFormByMatrix(mat,GF(5));
+Error, Invalid Gram matrix
+gap> STOP_TEST("bg_th_ex1.tst", 10000 );

tst/examples/bg_th_ex2.tst

@@ -0,0 +1,23 @@
+gap> START_TEST("Forms: bg_th_ex2.tst");
+gap> #Background theory: example 2
+gap> mat := [[1,0,0,0],[0,1,0,0],[0,0,1,0],[0,0,0,-1]]*Z(9)^0;
+[ [ Z(3)^0, 0*Z(3), 0*Z(3), 0*Z(3) ], [ 0*Z(3), Z(3)^0, 0*Z(3), 0*Z(3) ], 
+  [ 0*Z(3), 0*Z(3), Z(3)^0, 0*Z(3) ], [ 0*Z(3), 0*Z(3), 0*Z(3), Z(3) ] ]
+gap> form := BilinearFormByMatrix(mat,GF(9));
+< bilinear form >
+gap> Display(form);
+Bilinear form
+Gram Matrix:
+ 1 . . .
+ . 1 . .
+ . . 1 .
+ . . . 2
+gap> IsReflexiveForm(form);
+true
+gap> IsSymmetricForm(form);
+true
+gap> IsAlternatingForm(form);
+false
+gap> Dimension(RadicalOfForm(form));
+0
+gap> STOP_TEST("bg_th_ex2.tst", 10000 );

tst/examples/bg_th_ex3.tst

@@ -0,0 +1,20 @@
+gap> START_TEST("Forms: bg_th_ex3.tst");
+gap> #Background theory: example 3
+gap> mat := [[0,0,-2],[0,0,1],[2,-1,0]]*Z(7)^0;
+[ [ 0*Z(7), 0*Z(7), Z(7)^5 ], [ 0*Z(7), 0*Z(7), Z(7)^0 ], 
+  [ Z(7)^2, Z(7)^3, 0*Z(7) ] ]
+gap> form := BilinearFormByMatrix(mat,GF(7));
+< bilinear form >
+gap> Display(form);
+Bilinear form
+Gram Matrix:
+ . . 5
+ . . 1
+ 2 6 .
+gap> IsSymmetricForm(form);
+false
+gap> IsAlternatingForm(form);
+true
+gap> Dimension(RadicalOfForm(form));
+1
+gap> STOP_TEST("bg_th_ex3.tst", 10000 );

tst/examples/bg_th_ex4.tst

@@ -0,0 +1,30 @@
+gap> START_TEST("Forms: bg_th_ex4.tst");
+gap> #Background theory: example 4
+gap> mat := [[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,0,0,0,1],
+>         [0,0,0,0,1,0],[0,0,0,1,0,0],[0,0,1,0,0,0]]*Z(16)^0;
+[ [ 0*Z(2), Z(2)^0, 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2) ], 
+  [ Z(2)^0, 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2) ], 
+  [ 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2), Z(2)^0 ], 
+  [ 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2), Z(2)^0, 0*Z(2) ], 
+  [ 0*Z(2), 0*Z(2), 0*Z(2), Z(2)^0, 0*Z(2), 0*Z(2) ], 
+  [ 0*Z(2), 0*Z(2), Z(2)^0, 0*Z(2), 0*Z(2), 0*Z(2) ] ]
+gap> form := BilinearFormByMatrix(mat,GF(16));
+< bilinear form >
+gap> Display(form);
+Bilinear form
+Gram Matrix:
+ . 1 . . . .
+ 1 . . . . .
+ . . . . . 1
+ . . . . 1 .
+ . . . 1 . .
+ . . 1 . . .
+gap> IsSymmetricForm(form);
+true
+gap> IsAlternatingForm(form);
+true
+gap> IsDegenerateForm(form);
+false
+gap> WittIndex(form);
+3
+gap> STOP_TEST("bg_th_ex4.tst", 10000 );

tst/examples/bg_th_ex5.tst

@@ -0,0 +1,84 @@
+gap> START_TEST("Forms: bg_th_ex5.tst");
+gap> #Background theory: example 5
+gap> mat := [[0*Z(5),0*Z(5),0*Z(25),Z(25)^3],[0*Z(5),0*Z(5),Z(25)^3,0*Z(25)],
+>         [0*Z(5),-Z(25)^3,0*Z(5),0*Z(5)],[-Z(25)^3,0*Z(5),0*Z(25),0*Z(25)]];
+[ [ 0*Z(5), 0*Z(5), 0*Z(5), Z(5^2)^3 ], [ 0*Z(5), 0*Z(5), Z(5^2)^3, 0*Z(5) ], 
+  [ 0*Z(5), Z(5^2)^15, 0*Z(5), 0*Z(5) ], 
+  [ Z(5^2)^15, 0*Z(5), 0*Z(5), 0*Z(5) ] ]
+gap> form := HermitianFormByMatrix(mat,GF(25));
+< hermitian form >
+gap> Display(form);
+Hermitian form
+Gram Matrix:
+z = Z(25)
+    .    .    .  z^3
+    .    .  z^3    .
+    . z^15    .    .
+ z^15    .    .    .
+gap> WittIndex(form);
+2
+gap> form2 := BilinearFormByMatrix(mat,GF(25));
+< bilinear form >
+gap> Display(form2);
+Bilinear form
+Gram Matrix:
+z = Z(25)
+    .    .    .  z^3
+    .    .  z^3    .
+    . z^15    .    .
+ z^15    .    .    .
+gap> IsAlternatingForm(form2);
+true
+gap> Display(IsometricCanonicalForm(form));
+Hermitian form
+Gram Matrix:
+ 1 . . .
+ . 1 . .
+ . . 1 .
+ . . . 1
+Witt Index: 2
+gap> Display(IsometricCanonicalForm(form2));
+Bilinear form
+Gram Matrix:
+ . 1 . .
+ 4 . . .
+ . . . 1
+ . . 4 .
+Witt Index: 2
+gap> V := GF(25)^4;
+( GF(5^2)^4 )
+gap> u := [Z(5)^0,Z(5^2)^11,Z(5)^3,Z(5^2)^13 ];
+[ Z(5)^0, Z(5^2)^11, Z(5)^3, Z(5^2)^13 ]
+gap> [u,u]^form;
+0*Z(5)
+gap> v := [Z(5)^0,Z(5^2)^5,Z(5^2),Z(5^2)^13 ];
+[ Z(5)^0, Z(5^2)^5, Z(5^2), Z(5^2)^13 ]
+gap> [v,v]^form;                                     
+0*Z(5)
+gap> [u,v]^form;
+Z(5^2)^7
+gap> ([v,u]^form)^5;
+Z(5^2)^7
+gap> w := [Z(5^2)^21,Z(5^2)^19,Z(5^2)^4,Z(5)^3 ];
+[ Z(5^2)^21, Z(5^2)^19, Z(5^2)^4, Z(5)^3 ]
+gap> [w,w]^form;
+Z(5)
+gap> v := [Z(5)^0,Z(5^2)^10,Z(5^2)^15,Z(5^2)^3 ];
+[ Z(5)^0, Z(5^2)^10, Z(5^2)^15, Z(5^2)^3 ]
+gap> u := [Z(5)^3,Z(5^2)^9,Z(5^2)^4,Z(5^2)^16 ];
+[ Z(5)^3, Z(5^2)^9, Z(5^2)^4, Z(5^2)^16 ]
+gap> w := [Z(5)^2,Z(5^2)^9,Z(5^2)^23,Z(5^2)^11 ];
+[ Z(5)^2, Z(5^2)^9, Z(5^2)^23, Z(5^2)^11 ]
+gap> [u,v]^form;
+0*Z(5)
+gap> [u,w]^form;
+0*Z(5)
+gap> [v,w]^form;
+0*Z(5)
+gap> s := Subspace(V,[v,u,w]);
+<vector space over GF(5^2), with 3 generators>
+gap> Dimension(s);
+2
+gap> WittIndex(form);
+2
+gap> STOP_TEST("bg_th_ex5.tst", 10000 );

tst/examples/bg_th_ex6.tst

@@ -0,0 +1,20 @@
+gap> START_TEST("Forms: bg_th_ex6.tst");
+gap> #Background theory: example 6
+gap> V := GF(4)^3;                           
+( GF(2^2)^3 )
+gap> mat := [[Z(2^2)^2,Z(2^2),Z(2^2)^2],[Z(2^2)^2,Z(2)^0,Z(2)^0],
+>         [0*Z(2),Z(2)^0,0*Z(2)]];
+[ [ Z(2^2)^2, Z(2^2), Z(2^2)^2 ], [ Z(2^2)^2, Z(2)^0, Z(2)^0 ], 
+  [ 0*Z(2), Z(2)^0, 0*Z(2) ] ]
+gap> qform := QuadraticFormByMatrix(mat, GF(4));
+< quadratic form >
+gap> Display( qform );
+Quadratic form
+Gram Matrix:
+z = Z(4)
+ z^2   1 z^2
+   .   1   .
+   .   .   .
+gap> PolynomialOfForm( qform );
+Z(2^2)^2*x_1^2+x_1*x_2+Z(2^2)^2*x_1*x_3+x_2^2
+gap> STOP_TEST("bg_th_ex6.tst", 10000 );

tst/examples/bg_th_ex7.tst

@@ -0,0 +1,20 @@
+gap> START_TEST("Forms: bg_th_ex7.tst");
+gap> #Background theory: example 7
+gap> r := PolynomialRing(GF(8),4);
+GF(2^3)[x_1,x_2,x_3,x_4]
+gap> poly := r.1*r.2+r.3*r.4;
+x_1*x_2+x_3*x_4
+gap> qform := QuadraticFormByPolynomial(poly, r);
+< quadratic form >
+gap> Display(qform);
+Quadratic form
+Gram Matrix:
+ . 1 . .
+ . . . .
+ . . . 1
+ . . . .
+Polynomial: x_1*x_2+x_3*x_4
+
+gap> Dimension(RadicalOfForm(qform));
+0
+gap> STOP_TEST("bg_th_ex7.tst", 10000 );

tst/examples/bg_th_ex8.tst

@@ -0,0 +1,41 @@
+gap> START_TEST("Forms: bg_th_ex8.tst");
+gap> #Background theory: example 8
+gap> mat := [[Z(16)^3,1,0,0],[0,Z(16)^5,0,0],
+>              [0,0,Z(16)^3,1],[0,0,0,Z(16)^12]]*Z(16)^0;
+[ [ Z(2^4)^3, Z(2)^0, 0*Z(2), 0*Z(2) ], [ 0*Z(2), Z(2^2), 0*Z(2), 0*Z(2) ], 
+  [ 0*Z(2), 0*Z(2), Z(2^4)^3, Z(2)^0 ], [ 0*Z(2), 0*Z(2), 0*Z(2), Z(2^4)^12 ] 
+ ]
+gap> qform := QuadraticFormByMatrix(mat,GF(16));
+< quadratic form >
+gap> Display( qform );
+Quadratic form
+Gram Matrix:
+z = Z(16)
+  z^3    1    .    .
+    .  z^5    .    .
+    .    .  z^3    1
+    .    .    . z^12
+gap> mat2 := [[Z(16)^7,1,0,0],[0,0,0,0],
+>              [0,0,Z(16)^2,1],[0,0,0,Z(16)^9]]*Z(16)^0;
+[ [ Z(2^4)^7, Z(2)^0, 0*Z(2), 0*Z(2) ], [ 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2) ], 
+  [ 0*Z(2), 0*Z(2), Z(2^4)^2, Z(2)^0 ], [ 0*Z(2), 0*Z(2), 0*Z(2), Z(2^4)^9 ] ]
+gap> qform2 := QuadraticFormByMatrix(mat2, GF(16));
+< quadratic form >
+gap> Display( qform2 );
+Quadratic form
+Gram Matrix:
+z = Z(16)
+  z^7    1    .    .
+    .    .    .    .
+    .    .  z^2    1
+    .    .    .  z^9
+gap> biform := AssociatedBilinearForm( qform2 );
+< bilinear form >
+gap> Display( biform );
+Bilinear form
+Gram Matrix:
+ . 1 . .
+ 1 . . .
+ . . . 1
+ . . 1 .
+gap> STOP_TEST("bg_th_ex8.tst", 10000 );

tst/examples/bg_th_ex9.tst

@@ -0,0 +1,28 @@
+gap> START_TEST("Forms: bg_th_ex9.tst");
+gap> #Background theory: example 9
+gap> mat := [ [ Z(2^2), Z(2^2), Z(2^2), Z(2^2), Z(2^2) ], 
+>    [ 0*Z(2), Z(2^2), Z(2^2)^2, 0*Z(2), Z(2)^0 ], 
+>    [ 0*Z(2), 0*Z(2), Z(2)^0, Z(2)^0, Z(2)^0 ], 
+>    [ 0*Z(2), 0*Z(2), 0*Z(2), Z(2)^0, Z(2)^0 ], 
+>    [ 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2), Z(2)^0 ] ];;
+gap> qform := QuadraticFormByMatrix(mat,GF(4));
+< quadratic form >
+gap> IsSingularForm(qform);
+false
+gap> IsDegenerateForm(qform);
+#I  Testing degeneracy of the *associated bilinear form*
+true
+gap> biform := AssociatedBilinearForm(qform);
+< bilinear form >
+gap> Display(biform);
+Bilinear form
+Gram Matrix:
+z = Z(4)
+   . z^1 z^1 z^1 z^1
+ z^1   . z^2   .   1
+ z^1 z^2   .   1   1
+ z^1   .   1   .   1
+ z^1   1   1   1   .
+gap> IsDegenerateForm(biform);
+true
+gap> STOP_TEST("bg_th_ex9.tst", 10000 );