Testing Sign Conditions on a Multivariate Polynomial and Applications

Abstract.

Let f be a polynomial in ${\mathbb{Q}}[X_1, \ldots , X_n]$ 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 ≠ 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 $e \in {\mathbb{Q}}$ 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 $f : y \in {\mathbb{C}}^n \rightarrow f(y) \in {\mathbb{C}}$ 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 ${\mathcal{O}}(n^{7}D^{4n})$ arithmetic operations in ${\mathbb{Q}}$ . The paper ends with practical experiments showing the efficiency of our approach on real-life applications.