‣ PreparationForRadicalOfIdeal( I ) | ( attribute ) |
Returns: a LeftSubmodule
Computes the radical of an ideal if the coefficients field is perfect and otherwise computes the FGLM data of the ideal J which is generated by the separable parts of the minimal polynomials of the indeterminates evaluated for the indeterminates of the ring.
‣ RadicalOfIdeal( I ) | ( attribute ) |
Returns: a LeftSubmodule
Computes the radical of a zero dimensional ideal I.
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> radI := RadicalOfIdeal( I ); <A torsion-free ideal given by 6 generators> gap> Display( radI ); y*z-x, x*z-y, y^2-z^2, x*y-z, x^2-z^2, z^3-z An ideal generated by the 6 entries of the above matrix gap> IsSubset( I, radI ); true gap> A := HomalgRingOfIntegersInSingular( 3, "t" ) * "x,y"; GF(3)(t)[x,y] gap> J := LeftSubmodule( "x-t, y", A ); <A torsion-free ideal given by 2 generators> gap> rad := RadicalOfIdeal( J^2 ); <A torsion-free ideal given by 2 generators> gap> Display( rad ); y, -x+(t) An ideal generated by the 2 entries of the above matrix
‣ CompanionMatrix( mu ) | ( operation ) |
Returns: a matrix
Computes the companion matrix of the monic univariate polynomial mu over a univariate 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^2+x*y+z" / R; |[ x^2+x*y+z ]| gap> mu := MinimalPolynomial( r ); t^3-2*t^2-3*t gap> M := CompanionMatrix( mu ); <An unevaluated non-zero 3 x 3 matrix over an external ring> gap> Display( M ); 2,3,0, 1,0,0, 0,1,0
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