Quantitative curve selection lemma

Abstract

We prove a quantitative version of the curve selection lemma. Denoting by s, d, k bounds the number, the maximum total 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 in Jelonek and Kurdyka (Math Z 276:557–570, 2014). 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 in Le et al. (Computing the real isolated points of an algebraic hypersurface, 2020).