Extractions: Computable and Visible Analogues of Localizations for Polynomial Ideals
Abstract
When studying local properties of a polynomial ideal, one usually needs a theoretic technique called localization. For most cases, in spite of its importance, the computation in a localized ring cannot be algorithmically preformed. On the other hand, the standard basis method is very effective for the computation in a special kind of localized rings, but for a general semigroup order the geometry of the localization of a positive-dimensional ideal is difficult to interpret.
In this paper, we introduce a new ideal operation called extraction. For an ideal
I
in a polynomial ring
K[
x
1
,…,
x
n
]
over a field
K
, we use another ideal
J
to control the primary components of
I
and the result
β(I,J)
is called the extraction of
I
by
J
. It is still a polynomial ideal and has a concrete geometric meaning in
K
¯
n
, i.e., we keep the branches of
V(I)⊂
K
¯
n
that intersect with
V(J)⊂
K
¯
n
and delete others, where
K
¯
is the algebraic closure of
K
. This is what we mean by visible. On the other hand, we can use the standard basis method to compute a localized ideal corresponding to
β(I,J)
without a complete primary decomposition, and can do further computation in the localized ring such as determining the membership problem of
β(I,J)
. Moreover, we prove that extractions are as powerful as localizations in the sense that for any multiplicatively closed subset
S
of
K[
x
1
,…,
x
n
]
and any polynomial ideal
I
, there always exists a polynomial ideal
J
such that
β(I,J)=(
S
−1
I
)
c
.