‣ BasisOverCoefficientsRing( R ) | ( attribute ) |
Returns: a homalg matrix
Computes a basis of the ring R, provided it is free of finite rank over its coefficients ring.
gap> LoadPackage( "PrimaryDecomposition" ); true gap> Q := HomalgFieldOfRationalsInSingular( ); Q gap> A := Q * "x,y,z"; Q[x,y,z] gap> I := LeftSubmodule( "y*z-x, x*z-y, y^2-z^2, x*y-z, x^2-z^2, z^3-z", A ); <A torsion-free ideal given by 6 generators> gap> R := A / I; Q[x,y,z]/( y*z-x, x*z-y, y^2-z^2, x*y-z, x^2-z^2, z^3-z ) gap> bas := BasisOverCoefficientsRing( R ); <A 5 x 1 matrix over a residue class ring> gap> Display( bas ); 1, z, y, x, z^2 modulo [ y*z-x, x*z-y, y^2-z^2, x*y-z, x^2-z^2, z^3-z ]
‣ RepresentationOverCoefficientsRing( r ) | ( attribute ) |
Returns: a homalg matrix
Computes a representation matrix of the ring element r, with respect to the basis computed by BasisOverCoefficientsRing, provided the ring is free of finite rank over its coefficients ring.
gap> LoadPackage( "PrimaryDecomposition" ); true gap> Q := HomalgFieldOfRationalsInSingular( ); Q gap> A := Q * "x,y,z"; Q[x,y,z] gap> I := LeftSubmodule( "y*z-x, x*z-y, y^2-z^2, x*y-z, x^2-z^2, z^3-z", A ); <A torsion-free ideal given by 6 generators> gap> R := A / I; Q[x,y,z]/( y*z-x, x*z-y, y^2-z^2, x*y-z, x^2-z^2, z^3-z ) gap> r := HomalgRingElement( "x^2 +xy +z", R ); |[ x^2+x*y+z ]| gap> rep := RepresentationOverCoefficientsRing( r ); <A 5 x 5 matrix over an external ring> gap> Display( rep ); 0,2,0,0,1, 0,1,0,0,2, 0,0,1,2,0, 0,0,2,1,0, 0,2,0,0,1
‣ FGLMdata( R ) | ( attribute ) |
Returns: a list
Computes the FGLM data of the ring R (see [Jam13]), provided it is free of finite rank over its coefficients ring. The FGLM data of such a ring consists of the representation matrices of all indeterminates of R computed by RepresentationOverCoefficientsRing.
gap> LoadPackage( "PrimaryDecomposition" ); true gap> Q := HomalgFieldOfRationalsInSingular( ); Q gap> A := Q * "x,y,z"; Q[x,y,z] gap> I := LeftSubmodule( "y*z-x, x*z-y, y^2-z^2, x*y-z, x^2-z^2, z^3-z", A ); <A torsion-free ideal given by 6 generators> gap> R := A / I; Q[x,y,z]/( y*z-x, x*z-y, y^2-z^2, x*y-z, x^2-z^2, z^3-z ) gap> F:= FGLMdata(R); [ <A 5 x 5 matrix over an external ring>, <A 5 x 5 matrix over an external ring>, <A 5 x 5 matrix over an external ring> ] gap> Display(F[1]); 0,0,0,1,0, 0,0,1,0,0, 0,1,0,0,0, 0,0,0,0,1, 0,0,0,1,0 gap> Display(F[2]); 0,0,1,0,0, 0,0,0,1,0, 0,0,0,0,1, 0,1,0,0,0, 0,0,1,0,0 gap> Display(F[3]); 0,1,0,0,0, 0,0,0,0,1, 0,0,0,1,0, 0,0,1,0,0, 0,1,0,0,0
‣ MinimalPolynomial( r ) | ( attribute ) |
‣ MinimalPolynomial( r, str ) | ( operation ) |
‣ MinimalPolynomial( r, t ) | ( operation ) |
Returns: a matrix
Computes the minimal polynomial of the element r of a ring \(R\) of finite rank over its coefficients ring. The installed method computes the representation matrix \(M\) of the ring element r and finds the first linear dependency among the vectors \( e, eM, eM^2, ... \), where \(e\) is the identity of the ring.
gap> LoadPackage( "PrimaryDecomposition" ); true gap> Q := HomalgFieldOfRationalsInSingular( ); Q gap> A := Q * "x,y,z"; Q[x,y,z] gap> I := LeftSubmodule( "y*z-x, x*z-y, y^2-z^2, x*y-z, x^2-z^2, z^3-z", A ); <A torsion-free ideal given by 6 generators> gap> R := A / I; Q[x,y,z]/( y*z-x, x*z-y, y^2-z^2, x*y-z, x^2-z^2, z^3-z ) gap> r := "x^2+x*y+z" / R; |[ x^2+x*y+z ]| gap> s := "x*y+z"/ R; |[ x*y+z ]| gap> mu_r := MinimalPolynomial( r ); t^3-2*t^2-3*t gap> mu_s := MinimalPolynomial( s ); t^3-4*t gap> mu_r + mu_s; 2*t^3-2*t^2-7*t gap> kt := HomalgRing( mu_r ); Q[t] gap> t := Indeterminates( kt )[1]; t gap> mu := MinimalPolynomial( r, t ); t^3-2*t^2-3*t gap> mu = mu_r; true gap> mu := MinimalPolynomial( r, "t" ); t^3-2*t^2-3*t gap> mu = mu_r; true gap> mu := MinimalPolynomial( r, "a" ); a^3-2*a^2-3*a
‣ SquareFreeFactors( p ) | ( attribute ) |
Returns: a list
Computes the irreducible factors of the square free part of the univariate polynomial p.
gap> LoadPackage( "PrimaryDecomposition" ); true gap> A := HomalgFieldOfRationalsInSingular() * "t"; Q[t] gap> p := "t^3-2*t^2-3*t" / A; t^3-2*t^2-3*t gap> s := SquareFreeFactors( p ); [ t, t+1, t-3 ] gap> Product( s ); t^3-2*t^2-3*t
‣ SeparablePart( p ) | ( attribute ) |
Returns: a ring element
Computes the separable part of a univariate polynomial p. For polynomials over nonperfect rings the method uses an algorithm of Kemper (see [Kem91]).
gap> LoadPackage( "PrimaryDecomposition" ); true gap> A := HomalgFieldOfRationalsInSingular() * "t"; Q[t] gap> r := "t^4-t^3-5*t^2-3*t" / A; t^4-t^3-5*t^2-3*t gap> SquareFreeFactors( r ); [ t, t+1, t-3 ] gap> SeparablePart( r ); t^3-2*t^2-3*t gap> A := HomalgRingOfIntegersInSingular( 3, "t,s")* "x"; GF(3)(t,s)[x] gap> r := "( x^3 - s ) * ( x^3 + t )" / A; x^6+(t-s)*x^3+(-t*s) gap> SeparablePart( r ); x^2+(t-s)*x+(-t*s)
‣ BasisCoefficientsOfRingElement( r ) | ( attribute ) |
Returns: a matrix
Computes the coefficients of the basis representation of r, provided the ring is free of finite rank over its coefficients ring.
gap> LoadPackage( "PrimaryDecomposition" ); true gap> A := HomalgFieldOfRationalsInSingular( ) * "x,y,z"; Q[x,y,z] gap> I := LeftSubmodule( "y*z-x, x*z-y, y^2-z^2, x*y-z, x^2-z^2, z^3-z", A ); <A torsion-free ideal given by 6 generators> gap> R := A / I; Q[x,y,z]/( y*z-x, x*z-y, y^2-z^2, x*y-z, x^2-z^2, z^3-z ) gap> r := "x^4 * y^2 + z*x" / R; |[ x^4*y^2+x*z ]| gap> coeffs := BasisCoefficientsOfRingElement( r ); <A 1 x 5 matrix over an external ring> gap> Display( coeffs ); 0,0,1,0,1 gap> bas := BasisOverCoefficientsRing( R ); <A 5 x 1 matrix over a residue class ring> gap> Display( bas ); 1, z, y, x, z^2 modulo [ y*z-x, x*z-y, y^2-z^2, x*y-z, x^2-z^2, z^3-z ] gap> s := R * coeffs * bas ; <An unevaluated 1 x 1 matrix over a residue class ring> gap> Display( s ); z^2+y modulo [ y*z-x, x*z-y, y^2-z^2, x*y-z, x^2-z^2, z^3-z ]
‣ IdealBasisOverCoefficientRing( G ) | ( operation ) |
Returns: a matrix
Computes a \(K\)-basis of an ideal generated by the entries of the matrix G over \(R\), where \(K\) is the coefficient ring of \(R\), using an algorithm of Sebastian Jambor (see [Jam13]).
gap> LoadPackage( "PrimaryDecomposition" ); true gap> Q := HomalgFieldOfRationalsInSingular( ); Q gap> A := Q * "x,y,z"; Q[x,y,z] gap> I := LeftSubmodule( "y*z-x, x*z-y, y^2-z^2, x*y-z, x^2-z^2, z^3-z", A ); <A torsion-free ideal given by 6 generators> gap> R := A / I; Q[x,y,z]/( y*z-x, x*z-y, y^2-z^2, x*y-z, x^2-z^2, z^3-z ) gap> G := HomalgMatrix( "[ z + 1, y - 1 ]", 1, 1, R ); <A 1 x 1 matrix over a residue class ring> gap> B := IdealBasisOverCoefficientRing( G ); <An unevaluated 3 x 1 matrix over a residue class ring> gap> Display( B ); z+1, x+y, z^2+z modulo [ y*z-x, x*z-y, y^2-z^2, x*y-z, x^2-z^2, z^3-z ]
‣ GapInternalIsomorphicField( K ) | ( attribute ) |
Returns: a ring
Computes an isomorphic, GAP internal field, provided the field is perfect.
‣ IdealBasisToGroebner( M ) | ( attribute ) |
Returns: a list
Computes a Groebner basis of an ideal \( I \unlhd R \) defined by a \(K\)-basis of an ideal \(J \unlhd R/\tilde{I}\), where \(K\) is the coefficients ring and \(\tilde{I}\) a zero dimensional ideal. The basis elements are given in the rows of the matrix M in form of coefficient vectors with respect to the basis of the ring, which is computed by BasisOverCoefficientsRing and the matrix is defined over \(R/\tilde{I}\). This method works only for perfect coefficients rings \(K\), since the reduced echelon form only accepts GAP internal rings. The method is based on an algorithm of Sebastian Jambor (see [Jam13]).
‣ AppendToGroebnerBasisOfZeroDimensionalIdeal( G ) | ( attribute ) |
Returns: a matrix
Computes a Groebner Basis of the ideal generated by the entries of the matrix G and the generators of the defining zero dimensional ideal of the residue class ring over which G is defined. If the coefficients ring \(K\) is perfect, the method uses the IdealBasisOverCoefficientsRing and IdealBasisToGroebner. If not, the method uses the usual Groebner basis algorithms, since IdealBasisToGroebner only works for perfect coefficient rings.
gap> LoadPackage( "PrimaryDecomposition" ); true gap> A := HomalgFieldOfRationalsInSingular( ) * "x,y,z"; Q[x,y,z] gap> I := LeftSubmodule( "y*z-x, x*z-y, y^2-z^2, x*y-z, x^2-z^2, z^3-z", A ); <A torsion-free ideal given by 6 generators> gap> R := A/I; Q[x,y,z]/( y*z-x, x*z-y, y^2-z^2, x*y-z, x^2-z^2, z^3-z ) gap> J := HomalgMatrix( "[z]", 1, 1, R ); <A 1 x 1 matrix over a residue class ring> gap> mat := AppendToGroebnerBasisOfZeroDimensionalIdeal( J ); <An unevaluated 3 x 1 matrix over an external ring> gap> Display( mat ); z, y, x
‣ Derivative( f ) | ( attribute ) |
Returns: a ring element
Computes the derivative of the univariate polynomial f.
gap> LoadPackage( "PrimaryDecomposition" ); true gap> R := HomalgFieldOfRationalsInSingular() * "t"; Q[t] gap> r := "t^4-t^3-5*t^2-3*t" / R; t^4-t^3-5*t^2-3*t gap> Derivative( r ); 4*t^3-3*t^2-10*t-3 gap> S := HomalgRingOfIntegersInSingular( 3 ) * "t"; GF(3)[t] gap> r := "t^5 + 3*t^4 - t^3 + 2*t" / S; t^5-t^3-t gap> Derivative( r ); -t^4-1
‣ PolysOverTheSameRing( L ) | ( attribute ) |
Returns: a list
Accepts a list L of polynomials which are defined over function fields, whose coefficients rings are a root of the same function field. The method computes a polynomial ring over a function field and the representations of the polynomials in L over this ring. For each polynomial ring \(R_j\) containing L_j the number \(e_j = R_j\)!.RootOfBaseField defines the embedding map \(R \hookrightarrow R_j: a \mapsto a; t_i \mapsto t_i^{p^{e_j}}\) where \(p\) is the characteristic of the base field, \(a\) a field element and \(t_i\) are the rational parameters and \(R\) the common base function field. The rings \(R_j\) and \(R\) have to be technically identical, such that even the variable names coincide. The common ring \(S\) the method defines is in turn a copy of \(R\) with \(e_S = S\)!.RootOfBaseField is the least common multiply of the \(e_j\), excluding zero. The polynomials L_j will be represented over \(S\) by taking the \(t_i\) to the power of \(p^{e_S-e_j}\).
gap> LoadPackage( "PrimaryDecomposition" ); true gap> R1 := HomalgRingOfIntegersInSingular( 3, "t,s" ) * "x"; GF(3)(t,s)[x] gap> R1!.RootOfBaseField := 3; 3 gap> R2 := HomalgRingOfIntegersInSingular( 3, "t,s" ) * "x"; GF(3)(t,s)[x] gap> R2!.RootOfBaseField := 2; 2 gap> r1 := "(t*s)*x^3+(s^2)*x+(t)" / R1; (t*s)*x^3+(s^2)*x+(t) gap> r2 := "(t)*x^2+(s)" / R2; (t)*x^2+(s) gap> PolysOverTheSameRing( [ r1, r2 ] ); [ (t^27*s^27)*x^3+(s^54)*x+(t^27), (t^81)*x^2+(s^81) ] gap> R3 := HomalgRingOfIntegersInSingular( 3, "t,s" ) * "x"; GF(3)(t,s)[x] gap> R3!.RootOfBaseField := 0; 0 gap> r3 := "(t)*x^2+(-s)*x"/ R3; (t)*x^2+(-s)*x gap> PolysOverTheSameRing( [ r1, r2, r3 ] ); [ (t^27*s^27)*x^3+(s^54)*x+(t^27), (t^81)*x^2+(s^81), (t^729)*x^2+(-s^729)*x ]
‣ IsNotContainedInAnyHyperplane( lambda, L ) | ( operation ) |
Returns: a matrix
Determines whether lambda is contained in any \(n - 1 \) dimensional subspace spanned by the rows of L.
gap> LoadPackage( "PrimaryDecomposition" ); true gap> Q := HomalgFieldOfRationalsInSingular( ); Q gap> L := HomalgIdentityMatrix( 5, Q ); <An unevaluated 5 x 5 identity matrix over an external ring> gap> Display( L ); 1,0,0,0,0, 0,1,0,0,0, 0,0,1,0,0, 0,0,0,1,0, 0,0,0,0,1 gap> lambda := HomalgMatrix( "[ 2,0,2,1,1]", 1, 5, Q ); <A 1 x 5 matrix over an external ring> gap> Display( lambda ); 2,0,2,1,1 gap> IsNotContainedInAnyHyperplane( lambda, L ); false gap> lambda2 := GeneratorOfAnElementNotContainedInAnyHyperplane( L );; gap> IsNotContainedInAnyHyperplane( lambda2, L ); true
‣ GeneratorOfAnElementNotContainedInAnyHyperplane( L ) | ( operation ) |
‣ GeneratorOfAnElementNotContainedInAnyHyperplane( L, C ) | ( operation ) |
Returns: a value
Generates an element of \(K^n\), which is not contained in any \(n - 1\) dimensional subspace of the vector field \(K^n\), respectively \(C^n\), spanned by the rows of L. The method repeats generating a random vector \(\lambda \in K^n\) until the method IsNotContainedInAnyHyperplan of \(\lambda\) and L returns true.
gap> LoadPackage( "PrimaryDecomposition" ); true gap> Q := HomalgFieldOfRationalsInSingular( ); Q gap> e := CertainRows( HomalgIdentityMatrix( 5, Q ), [ 1 .. 3 ] ); <An unevaluated diagonal right invertible sub-identity 3 x 5 matrix over an external ring> gap> lambda := GeneratorOfAnElementNotContainedInAnyHyperplane( e ); <A non-zero left regular 1 x 5 matrix over an external ring> gap> l := UnionOfRows( e, lambda ); <An unevaluated non-zero 4 x 5 matrix over an external ring> gap> lambda := GeneratorOfAnElementNotContainedInAnyHyperplane( l ); <A non-zero left regular 1 x 5 matrix over an external ring> gap> A := HomalgRingOfIntegersInSingular( 3, "t" ); GF(3)(t) gap> S := HomalgRingOfIntegersInSingular( 3, "t" ); GF(3)(t) gap> S!.RootOfBaseField:= 1; 1 gap> L := HomalgMatrix( "[ 2, t, 0, t, 0, 1 ]", 2, 3, A ); <A 2 x 3 matrix over an external ring> gap> GeneratorOfAnElementNotContainedInAnyHyperplane( L, S ); <A 1 x 3 matrix over an external ring>
‣ FGLMToGroebner( M, e ) | ( operation ) |
‣ FGLMToGroebner( M, e, l ) | ( operation ) |
Returns: two lists
Computes a basis of the residue class ring and a Groebner Basis of the defining ideal, both defined by the FGLM data M and the identity element e. The method uses an algorithm of Sebastian Jambor (see [Jam13]).
gap> LoadPackage( "PrimaryDecomposition" ); true gap> A := HomalgFieldOfRationalsInSingular( ) * "x1,x2,x3"; Q[x1,x2,x3] gap> I := LeftSubmodule( "x2*x3-x1, x1*x3-x2, x2^2-x3^2, x1*x2-x3, \ > x1^2-x3^2, x3^3-x3", A ); <A torsion-free ideal given by 6 generators> gap> R := A / I; Q[x1,x2,x3]/( x2*x3-x1, x1*x3-x2, x2^2-x3^2, x1*x2-x3, x1^2-x3^2, x3^3-x3 ) gap> M := FGLMdata( R ); [ <A 5 x 5 matrix over an external ring>, <A 5 x 5 matrix over an external ring>, <A 5 x 5 matrix over an external ring> ] gap> e := CertainRows( > HomalgIdentityMatrix( NrRows( M[1] ), HomalgRing( M[1] ) ), [1] ); <An unevaluated diagonal right invertible sub-identity 1 x 5 matrix over an external ring> gap> FGLMToGroebner( M, e ); [ [ 1, x3, x2, x1, x3^2 ], [ -x2*x3+x1, -x2^2+x3^2, -x1*x3+x2, -x1*x2+x3, -x1^2+x3^2, -x3^3+x3 ] ] gap> bas := BasisOverCoefficientsRing( R ); <A 5 x 1 matrix over a residue class ring> gap> Display( bas ); 1, x3, x2, x1, x3^2 modulo [ x2*x3-x1, x1*x3-x2, x2^2-x3^2, x1*x2-x3, x1^2-x3^2, x3^3-x3 ]
‣ MatrixEmbedding( M, f ) | ( operation ) |
Returns: a matrix
Computes the image of the matrix \(\textit{M}\in K(\alpha)^{r \times c})\) where \(K(\alpha)\) with minimal polynomial f, under the extension field \(K(\alpha)\) with minimal polynomial f embedding \(K(\alpha)^{r \times c } \hookrightarrow K^{r deg(\textit{f}) \times c deg(\textit{f})}\) which is a the natural extension of \(K(\alpha )\hookrightarrow K^{deg(\textit{f})\times deg(\textit{f})}: \alpha \mapsto \texttt{CompanionMatrix}(\alpha)\). The method provides that the generator of the field extension \(\alpha\) coincides withe the negativ of the constant coefficients of the minimal polynomial.
gap> LoadPackage( "PrimaryDecomposition" ); true gap> R := HomalgRingOfIntegersInSingular( 3, "t,s" ) * "x"; GF(3)(t,s)[x] gap> R!.RootOfBaseField:=1; 1 gap> f := "x^3 - t" / R; x^3+(-t) gap> p := "x^3 - s" / R; x^3+(-s) gap> A := CoefficientsRing( R ); GF(3)(t,s) gap> M := HomalgMatrix( "[t*s - s, 1, 0, s]", 2, 2, A ); <A 2 x 2 matrix over an external ring> gap> Display( M ); (t*s-s),1, 0, (s) gap> N := MatrixEmbedding( M, f ); <An unevaluated non-zero 6 x 6 matrix over an external ring> gap> Display( N ); (-s),0, (t*s),1, 0, 0, (s), (-s),0, 0, 1, 0, 0, (s), (-s), 0, 0, 1, 0, 0, 0, (s),0, 0, 0, 0, 0, 0, (s),0, 0, 0, 0, 0, 0, (s) gap> S := HomalgRingOfIntegersInSingular( 3, "t" ) * "x"; GF(3)(t)[x] gap> S!.RootOfBaseRing := 3; 3 gap> I := LeftSubmodule( "x^2 - t^3", S ); <A principal torsion-free ideal given by a cyclic generator> gap> L := S / I; GF(3)(t)[x]/( x^2+(-t^3) ) gap> FGLM := FGLMdata( L ); [ <A 2 x 2 matrix over an external ring> ] gap> Display( FGLM[1] ); 0, 1, (t^3),0 gap> f := "x^3 - t" / S; x^3+(-t) gap> M := MatrixEmbedding( FGLM[1] , f ); <An unevaluated non-zero 6 x 6 matrix over an external ring> gap> Display( M ); 0, 0, 0, 1,0,0, 0, 0, 0, 0,1,0, 0, 0, 0, 0,0,1, (t),0, 0, 0,0,0, 0, (t),0, 0,0,0, 0, 0, (t),0,0,0 gap> T := HomalgRing( M ); GF(3)(t) gap> e := CertainRows( HomalgIdentityMatrix( NrRows( M ), T ), [1] ); <An unevaluated diagonal right invertible sub-identity 1 x 6 matrix over an external ring> gap> FGLMToGroebner( [ M ], e ); [ [ 1, x ], [ -x^2+(t) ] ]
‣ CoefficientsTransformation( M, deg ) | ( operation ) |
Returns: a matrix
Raises the rational parameters to the power \(p\)^deg in the entries of the matrix M, where p is the characteristic of the base field.
gap> LoadPackage( "PrimaryDecomposition" ); true gap> C := HomalgRingOfIntegersInSingular( 3, "t,s" ); GF(3)(t,s) gap> deg := 2; 2 gap> M := HomalgMatrix( "[ t, s*t, s^2, 1 ]", 4, 1, C ); <A 4 x 1 matrix over an external ring> gap> Display( M ); (t), (t*s), (s^2), 1 gap> coeffs := CoefficientsTransformation( M, deg ); <A 4 x 1 matrix over an external ring> gap> Display( coeffs ); (t^9), (t^9*s^9), (s^18), 1
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