An Algorithm for Computing the Limit Points of the Quasi-component of a Regular Chain

TL;DR

This paper introduces an algorithm that computes the limit points of the quasi-component of a regular chain using Puiseux series, avoiding the need for generators of the saturated ideal, with promising experimental results.

Contribution

The paper presents a novel algorithm for computing limit points of regular chain quasi-components without requiring generators of the saturated ideal, focusing on dimension one cases.

Findings

Algorithm effectively computes limit points in dimension one cases.

Experimental results demonstrate efficiency and benefits of the proposed method.

Potential extensions to higher-dimensional cases are discussed.

Abstract

For a regular chain $R$, we propose an algorithm which computes the (non-trivial) limit points of the quasi-component of $R$, that is, the set $\bar{W(R)} \setminus W(R)$. Our procedure relies on Puiseux series expansions and does not require to compute a system of generators of the saturated ideal of $R$. We focus on the case where this saturated ideal has dimension one and we discuss extensions of this work in higher dimensions. We provide experimental results illustrating the benefits of our algorithms.