Testing Sign Conditions on a Multivariate Polynomial and Applications

Let $f$ be a polynomial in $Q[X_{1},…,X_{n}]$ of degree $D$. We focus on testing the emptiness and computing at least one point in each connected component of the semi-algebraic set defined by $f > 0$ (or $f < 0$ or $f \neq 0$). To this end, the problem is reduced to computing at least one point in each connected component of a hypersurface defined by $f − e = 0$ for $e \in Q$ positive and small enough. We provide an algorithm allowing us to determine a positive rational number e which is small enough in this sense. This is based on the efficient computation of the set of generalized critical values of the mapping $f:y \in C^{n} \to f(y) \in C$ which is the union of the classical set of critical values of the mapping $f$ and the set of asymptotic critical values of the mapping $f$. Then, we show how to use the computation of generalized critical values in order to obtain an efficient algorithm deciding the emptiness of a semi-algebraic set defined by a single inequality or a single inequation. At last, we show how to apply our contribution to determining if a hypersurface contains real regular points. We provide complexity estimates for probabilistic versions of the latter algorithms which are within $O(n^{7}D^{4n})$ arithmetic operations in $Q$. The paper ends with practical experiments showing the efficiency of our approach on real-life applications.