Toric Border Bases

We extend the theory and the algorithms of Border Bases to systems of Laurent polynomial equations, defining "toric" roots. Instead of introducing new variables and new relations to saturate by the variable inverses, we propose a more efficient approach which works directly with the variables and their inverse. We show that the commutation relations and the inversion relations characterize toric border bases. We explicitly describe the first syzygy module associated to a toric border basis in terms of these relations. Finally, a new border basis algorithm for Laurent polynomials is described and a proof of its termination is given for zero-dimensional toric ideals

Strong bi-homogeneous B\'{e}zout theorem and its use in effective real algebraic geometry

Let f1, ..., fs be a polynomial family in Q[X1,..., Xn] (with s less than n) of degree bounded by D. Suppose that f1, ..., fs generates a radical ideal, and defines a smooth algebraic variety V. Consider a projection P. We prove that the degree of the critical locus of P restricted to V is bounded by D^s(D-1)^(n-s) times binomial of n and n-s. This result is obtained in two steps. First the critical points of P restricted to V are characterized as projections of the solutions of Lagrange's system for which a bi-homogeneous structure is exhibited. Secondly we prove a bi-homogeneous B\'ezout Theorem, which bounds the sum of the degrees of the equidimensional components of the radical of an ideal generated by a bi-homogeneous polynomial family. This result is improved when f1,..., fs is a regular sequence. Moreover, we use Lagrange's system to design an algorithm computing at least one point in each connected component of a smooth real algebraic set. This algorithm generalizes, to the non equidimensional case, the one of Safey El Din and Schost. The evaluation of the output size of this algorithm gives new upper bounds on the first Betti number of a smooth real algebraic set. Finally, we estimate its arithmetic complexity and prove that in the worst cases it is polynomial in n, s, D^s(D-1)^(n-s) and the binomial of n and n-s, and the complexity of evaluation of f1,..., fs

Border basis representation of a general quotient algebra

International audienceIn this paper, we generalized the construction of border bases to non-zero dimensional ideals for normal forms compatible with the degree, tackling the remaining obstacle for a general application of border basis methods. First, we give conditions to have a border basis up to a given degree. Next, we describe a new stopping criteria to determine when the reduction with respect to the leading terms is a normal form. This test based on the persistence and regularity theorems of Gotzmann yields a new algorithm for computing a border basis of any ideal, which proceeds incrementally degree by degree until its regularity. We detail it, prove its correctness, present its implementation and report some experimentations which illustrate its practical good behavior

Circular Cylinders by Four or Five Points in Space

International audienceWe are interested in computing effectively cylinders through 5 points, and in other problems involved in metrology. In particular, we consider the cylinders through 4 points with a fix radius and with extremal radius. For these different problems, we give bounds on the number of solutions and exemples show that these bounds are optimal. Finally, we describe two algebraic methods which can be used here to solve efficiently these problems and some experimentation results

Vers une résolution stable et rapide des équations algébriques

PARIS-BIUSJ-Thèses (751052125) / SudocPARIS-BIUSJ-Mathématiques rech (751052111) / SudocSudocFranceF

Generalized normal forms and polynomial system solving

International audienceThis report describes a new method for computing the normal form of a polynomial modulo a zero-dimensional ideal $I$. We give a detailed description of the algorithm, a proof of its correctness, and finally experimentations on classical benchmark polynomial systems. The method that we propose can be thought as an extension of both the Gröbner basis method and the Macaulay construction. As such it establishes a natural link between these two methods. We have weaken the monomial ordering requirement for Gröbner bases computations, which allows us to construct new type of representations for the associated quotient algebra. This approach yields more freedom in the linear algebra steps involved, which allows us to take into account numerical criteria while performing the symbolic steps. This is a new feature for a symbolic algorithm, which has an important impact on the practical efficiency, as it is illustrated by the experiments at the end of the paper

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

Let $(f_1, \ldots, f_s)$ be a polynomial family in \Q[X_1, \ldots, X_n] (with $s\leq n-1$) of degree bounded by $D$, generating a radical ideal, and defining a smooth algebraic variety \mathcal{V}\subset\C Consider a {\em generic} projection \pi:\Cightarrow\Cts restriction to $\mathcal{V}$ and its critical locus which is supposed to be zero-dimensional. We state that the number of critical points of $\pi$ restricted to $\mathcal{V}$ is bounded by $D^s(D-1)^{n-s}{{n}\choose{n-s}}$. This result is obtained in two steps. First the critical points of $\pi$ restricted to $\mathcal{V}$ are characterized as projections of the solutions of the Lagrange system for which a bi-homogeneous structure is exhibited. Secondly we apply a 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 $(f_1, \ldots, f_s)$ 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, evaluating the size of the output of this algorithm gives new upper bounds on the first Betti number of a smooth real algebraic set

Strong bihomogeneous Bézout theorem and degree bounds for algebraic optimization

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

UNCONSTRAINT GLOBAL POLYNOMIAL OPTIMIZATION VIA GRADIENT IDEAL

Abstract. In this paper, we describe a new method to compute the minimum of a real polynomial function and the ideal defining the points which minimize this polynomial func-tion, assuming that the minimizer ideal is zero-dimensional. Our method is a generalization of Lasserre relaxation method and stops in a finite number of steps. The proposed algo-rithm combines Border Basis, Moment Matrices and Semidefinite Programming. In the case where the minimum is reached at a finite number of points, it provides a border basis of the minimizer ideal. 1