Puiseux expansions and non-isolated points in algebraic varieties

TL;DR

This paper develops criteria using Puiseux series to determine whether solutions to polynomial systems are isolated or form continuous curves, enhancing understanding of solution set structures in algebraic geometry.

Contribution

It introduces new conditions based on truncated Puiseux series to identify non-isolated points and partial parametrizations in algebraic varieties.

Findings

Criteria for non-isolated solutions using Puiseux series

Conditions for partial curve parametrizations in 1-dimensional solution sets

Enhanced methods for analyzing solution set structures

Abstract

We consider the problem of deciding whether a common solution to a multivariate polynomial equation system is isolated or not. We present conditions on a given truncated Puiseux series vector centered at the point ensuring that it is not isolated. In addition, in the case that the set of all common solutions of the system has dimension 1, we obtain further conditions specifying to what extent the given vector of truncated Puiseux series coincides with the initial part of a parametrization of a curve of solutions passing through the point.