Quelle Eliminate.g
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## <#GAPDoc Label="Eliminate">
## <Subsection Label="Eliminate">
## <Heading>Eliminate</Heading>
## <Example><![CDATA[
## gap> R := HomalgFieldOfRationalsInDefaultCAS( ) * "x,y,z,l,m";
## Q[x,y,z,l,m]
## gap> var := Indeterminates( R );
## [ x, y, z, l, m ]
## gap> x := var[1];; y := var[2];; z := var[3];; l := var[4];; m := var[5];;
## gap> L := [ x*m+l-4, y*m+l-2, z*m-l+1, x^2+y^2+z^2-1, x+y-z ];
## [ x*m+l-4, y*m+l-2, z*m-l+1, x^2+y^2+z^2-1, x+y-z ]
## gap> e := Eliminate( L, [ l, m ] );
## <A non-zero right regular 3 x 1 matrix over an external ring>
## gap> Display( e );
## 4*y+z,
## 4*x-5*z,
## 21*z^2-8
## gap> I := LeftSubmodule( e );
## <A torsion-free (left) ideal given by 3 generators>
## gap> Display( I );
## 4*y+z,
## 4*x-5*z,
## 21*z^2-8
##
## A (left) ideal generated by the 3 entries of the above matrix
## gap> J := LeftSubmodule( "x+y-z, -2*z-3*y+x, x^2+y^2+z^2-1", R );
## <A torsion-free (left) ideal given by 3 generators>
## gap> I = J;
## true
## ]]></Example>
## </Subsection>
## <#/GAPDoc>
LoadPackage( "RingsForHomalg" );
R := HomalgFieldOfRationalsInDefaultCAS( ) * "x,y,z,l,m";
var := Indeterminates( R );
x := var[1]; y := var[2]; z := var[3]; l := var[4]; m := var[5];
L := [ x*m+l-4, y*m+l-2, z*m-l+1, x^2+y^2+z^2-1, x+y-z ];
e := Eliminate( L, [ l, m ] );
LoadPackage( "Modules" );
I := LeftSubmodule( e );
J := LeftSubmodule( "x+y-z, -2*z-3*y+x, x^2+y^2+z^2-1", R );
Assert( 0, I = J );
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2026-03-28
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