Let f be a polynomial in Q[X1,..., Xn] 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 ∈ 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 ∈ C n → f(y) ∈ 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 semialgebraic 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.

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We have developed in the past several algorithms with intrinsic complexity bounds for the problem of point finding in real algebraic varieties. Our aim here is to give a comprehensive presentation of the geometrical tools which are necessary to prove the correctness and complexity estimates of these algorithms. Our results form also the geometrical main ingredients for the computational treatment of singular hypersurfaces. In particular, we show the non–emptiness of suitable generic dual polar varieties of (possibly singular) real varieties, show that generic polar varieties may become singular at smooth points of the original variety and exhibit a sufficient criterion when this is not the case. Further, we introduce the new concept of meagerly generic polar varieties and give a degree estimate for them in terms of the degrees of generic polar varieties. The statements are illustrated by examples and a computer experiment.

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We describe an algorithm (VQE) for a variant of the real quantifier elimination problem (QE). The variant problem requires the input to satisfy a certain extra condition, and allows the output to be almost equivalent to the input. The motivation/rationale for studying such a variant QE problem is that many quantified formulas arising in applications do satisfy the extra conditions. Furthermore, in most applications, it is sufficient that the output formula is almost equivalent to the input formula. The main idea underlying the algorithm is to substitute the repeated projection step of CAD by a single projection without carrying out a parametric existential decision over the reals. We find that the algorithm can tackle important and challenging problems, such as numerical stability analysis of the widely-used MacCormack’s scheme. The problem has been practically out of reach for standard QE algorithms in spite of many attempts to tackle it. However the current implementation of VQE can solve it in about 12 hours. This paper extends the results reported at the conference ISSAC 2009.

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We study a variant of the real quantifier elimination problem (QE). The variant problem requires the input to satisfy a certain extra condition, and allows the ouput to be almost equivalent to the input. In a sense, we are strengthening the pre-condition and weakening the post-condition of the standard QE problem. The motivation/rationale for studying such a variant QE problem is that many quantified formulas arising in applications do satisfy the extra conditions. Furthermore, in most applications, it is sufficient that the ouput formula is almost equivalent to the input formula. Thus, we propose to solve a variant of the initial quantifier elimination problem. We present an algorithm (VQE) , that exploits the strengthened pre-condition and the weakened post-condition. The main idea underlying the algorithm is to substitute the repeated projection step of CAD by a single projection without carrying out a parametric existential decision over the reals. We find that the algorithm VQE can tackle important and challenging problems, such as numerical stability analysis of the widely-used MacCormack’s scheme. The problem has been practically out of reach for standard QE algorithms in spite of many attempts to tackle it. However the current implementation of VQE can solve it in about 1 day.

Let (f1,..., fs) be a polynomial family in Q[X1,..., Xn] (with s ≤ n − 1) of degree bounded by D, generating a radical ideal, and defining a smooth algebraic variety V ⊂ C n. Consider a generic projection π: C n → C, its restriction to V and its critical locus which is supposed to be zero-dimensional. We state that the number of critical points of π restricted to V is bounded by D s (D−1) n−s � n n−s �. This result is obtained in two steps. First the critical points of π restricted to V are characterized as projections of the solutions of the Lagrange system for which a bi-homogeneous structure is exhibited. Next, we apply a strong bi-homogeneous Bézout Theorem, for which we give a proof and which bounds the sum of the degrees of the isolated primary components of an ideal generated by a bi-homogeneous family for which we give a proof. This result is improved in the case where (f1,..., fs) is a regular sequence. Moreover, we use Lagrange’s system to generalize the algorithm due to Safey El Din and Schost for computing at least one point in each connected component of a smooth real algebraic set to the non equidimensional case. Then, the evaluation of the size of the output of this algorithm gives new upper bounds on the first Betti number of a smooth real algebraic set.

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