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Initial ideal

InitialIdeal(I: IDEAL, Inds: LIST): IDEAL

Let Inds be a subset of the set of indeterminates, and let 0 be the degree of the remaining indeterminates. The initial form with respect to Inds of a polynomial f is the homogeneous component of f of the lowest degree (in contrast with the leading form, see LF , DF ). The initial ideal of the ideal I is the ideal generated by the initial forms of all polynomials in I .

If Inds is the set of all indeterminates then the initial ideal is also called the tangent cone of I ( TgCone ).

The implementation is based on Lazard's method (see Kreuzer-Robbiano, Computational Commutative Algebra 2, pg.463).

/**/  Use R ::= QQ[x,y];
/**/  I := ideal(x^3 +x^2 -y^2);
/**/  InitialIdeal(I, [x,y]);
ideal(x^2 -y^2)
/**/  TgCone(I);
ideal(x^2 -y^2)

/**/  Use R ::= QQ[x,y];
/**/  I := ideal(x^2 +x*y);
/**/  InitialIdeal(I, [x,y]);
ideal(x^2 +x*y)
/**/  InitialIdeal(I, [x]);

See Also