‣ 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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