Quantitative Curve Selection Lemma

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 $s^{2k+1}{d}^{O(k)}$ improving a result of Le, Safey el Din, and de Wolff.