Extractions: Computable and Visible Analogues of Localizations for Polynomial Ideals

TL;DR

This paper introduces the extraction operation for polynomial ideals, providing a computable, geometric analogue of localization that simplifies analyzing local properties without full primary decomposition.

Contribution

It defines extraction as a new ideal operation that captures local geometric features and can be computed using standard basis methods, bridging the gap between algebraic and geometric localizations.

Findings

Extraction retains geometric meaning in polynomial rings.

Extraction can be computed without primary decomposition.

Extraction is as powerful as localization 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 $\beta(I,J)=(S^{-1}I)^c$.