In this paper, we present a new method for solving square polynomial systems with no zero at infinity. We analyze its complexity, which indicates substantial improvements, compared with the previously known methods for solving such systems. We describe a framework for symbolic and numeric computations, developed in C++, in which we have implemented this algorithm. We mention the techniques that are involved in order to build efficient codes and compare with existing softwares. We end by some applications of this method, considering in particular an autocalibration problem in Computer Vision and an identification problem in Signal Processing, and report on the results of our first implementation.

Let k be a eld of characteristic zero. We present a fast algorithm for multiplying multivariate power series over k truncated in total degree. Up to logarithmic factors, its complexity is optimal, i.e. linear in the number of coecients of the series. 1.

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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Recent theoretical advances in elimination theory use straight-line programs as a datastructure to represent multivariate polynomials. We present here the Projective Noether Package which is a Maple implementation of one of these new algorithms, yielding as a byproduct a computation of the dimension of a projective variety. Comparative results on benchmarks for time and space of several families of multivariate polynomial equation systems are given and we point out both weaknesses and advantages of different approaches.

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.