Quantitative Curve Selection Lemma

TL;DR

This paper presents a quantitative version of the curve selection lemma, providing explicit bounds on the degree of semi-algebraic paths and their Zariski closures, with applications to describing real isolated points.

Contribution

It introduces a quantitative curve selection lemma with explicit degree bounds and an improved algorithm for describing real isolated points in semi-algebraic sets.

Findings

Degree bounds for semi-algebraic paths are explicitly quantified.

Improved bounds on the degree of Zariski closures of paths.

Enhanced algorithm for identifying real isolated points with better complexity.

Abstract

We prove a quantitative version of the curve selection lemma. Denoting by $s,d,k$ a bound on the number, the degree and the number of variables of the polynomials describing a semi-algebraic set $S$ and a point $x$ in $\bar S$, we find a semi-algebraic path starting at $x$ and entering in $S$ with a description of degree $(O(d)^{3k+3},O(d)^{k})$ (using a precise definition of the description of a semi-algebraic path and its degree given in the paper). As a consequence, we prove that there exists a semi-algebraic path starting at $x$ and entering in $S$, such that the degree of the Zariski closure of the image of this path is bounded by $O(d)^{4k+3}$, improving a result of Jelonek and Kurdyka. We also give an algorithm for describing the real isolated points of $S$ whose complexity is bounded by $s^{2 k+1}d^{O(k)}$ improving a result of Le, Safey el Din, and de Wolff.