Mohab Safey El Din

Real solving for positive dimensional systems

Finding one point on each semi-algebraically connected component of a real algebraic variety, or at least deciding if such a variety is empty or not, is a fundamental problem of computational real algebraic geometry. Although numerous studies have been done on the subject, only a small number of efficient implementations exist.In this paper, we propose a new efficient and practical algorithm for computing such points. By studying the critical points of the restriction to the variety of the distance function to one well chosen point, we show how to provide a set of zero-dimensional systems whose zeros contain at least one point on each semi-algebraically connected component of the studied variety, without any assumption either on the variety (smoothness or compactness for example) or on the system of equations which define it.From the output of our algorithm, one can then apply, for each computed zero-dimensional system, any symbolic or numerical algorithm for counting or approximating the real solutions. We report some experiments using a set of pure exact methods. The practical efficiency of our method is due to the fact that we do not apply any infinitesimal deformations, unlike the existing methods based on a similar strategy.

On the complexity of computing gröbner bases for quasi-homogeneous systems

Let <i>K</i> be a field and (<i>f</i>₁, ..., <i>f_n</i>)subset <i>K</i>[<i>X</i>₁, ..., <i>X_n</i>] be a sequence of quasi-homogeneous polynomials of respective weighted degrees (<i>d</i>₁, ..., <i>d_n</i>) w.r.t a system of weights (<i>w</i>₁,...,<i>w_n</i>). Such systems are likely to arise from a lot of applications, including physics or cryptography.n We design strategies for computing Gröbner bases for quasi-homogeneous systems by adapting existing algorithms for homogeneous systems to the quasi-homogeneous case. Overall, under genericity assumptions, we show that for a generic zero-dimensional quasi homogeneous system, the complexity of the full strategy is polynomial in the weighted Bézout bound Π_{<i>i</i>=1^<i>n</i>}<i>d</i>ⁱ / Π_{<i>i</i>=1^<i>n</i><i>wⁱ</i>.n We provide some experimental results based on generic systems as well as systems arising from a cryptography problem. They show that taking advantage of the quasi-homogeneous structure of the systems allow us to solve systems that were out of reach otherwise.

Variant real quantifier elimination: algorithm and application

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.n 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.n 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.n We find that the algorithm VQE can tackle important and challenging problems, such as numerical stability analysis of the widely-used MacCormacks 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.

Critical point methods and effective real algebraic geometry: new results and trends

Critical point methods are at the core of the interplay between polynomial optimization and polynomial system solving over the reals. These methods are used in algorithms for solving various problems such as deciding the existence of real solutions of polynomial systems, performing one-block real quantifier elimination, computing the real dimension of the solution set, etc. The input consists of

On the practical computation of one point in each connected component of a semi-algebraic set defined by a polynomial system of equations and non-strict inequalities

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Solvers for ALgebraic Systems and Applications

polynomials in

Computing Gröbner bases for quasi-homogeneous systems

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Strong Bi-homogeneous Bézout's Theorem and degree bounds for algebraic optimization

variables of degree at most

Solving Problems through Algebraic Computation and Efficient Software

D

De l'algèbre linéaire à la résolution des systèmes polynomiaux

. Usually, the complexity of the algorithms is