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#### 4.1 Attributes

 ‣ 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>
<A torsion-free ideal given by 6 generators>
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
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>
<A torsion-free ideal given by 2 generators>
y,
-x+(t)

An ideal generated by the 2 entries of the above matrix


##### 4.1-3 CompanionMatrix
 ‣ 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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