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InitialIdeal --
Initial ideal
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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 the method of Lazard
(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]);
ideal(x*y)
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