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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]);
ideal(x*y)
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