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 $K[x_1,\ldots,x_n]$ over a field $K$, we use another ideal $J$ to control the primary components of $I$ and the result $\beta(I,J)$ is called the extraction of $I$ by $J$. It is still a polynomial ideal and has a concrete geometric meaning in $\bar{K}^n$, i.e., we keep the branches of $\textbf{V}(I) \subset \bar{K}^n$ that intersect with $\textbf{V}(J) \subset \bar{K}^n$ and delete others, where $\bar{K}$ 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 $\beta(I,J)$ without a complete primary decomposition, and can do further computation in the localized ring such as determining the membership problem of $\beta(I,J)$. Moreover, we prove that extractions are as powerful as localizations in the sense that for any multiplicatively closed subset $S$ of $K[x_1,\ldots,x_n]$ and any polynomial ideal $I$, there always exists a polynomial ideal $J$ such that $\beta(I,J)=(S^{-1}I)^c$.