‣ 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 ⊴ R defined by a K-basis of an ideal J ⊴ R/tildeI, where K is the coefficients ring and tildeI 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/tildeI. 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 ↪ R_j: a ↦ a; t_i ↦ 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 λ ∈ K^n until the method IsNotContainedInAnyHyperplan of λ 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 M∈ K(α)^r × c) where K(α) with minimal polynomial f, under the extension field K(α) with minimal polynomial f embedding K(α)^r × c ↪ K^r deg(f) × c deg(f) which is a the natural extension of K(α )↪ K^deg(f)× deg(f): α ↦ textttCompanionMatrix(α). The method provides that the generator of the field extension α 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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