VPSPACE and a transfer theorem over the complex field

We extend the transfer theorem of [KP2007] to the complex field. That is, we investigate the links between the class VPSPACE of families of polynomials and the Blum-Shub-Smale model of computation over C. Roughly speaking, a family of polynomials is in VPSPACE if its coefficients can be computed in polynomial space. Our main result is that if (uniform, constant-free) VPSPACE families can be evaluated efficiently then the class PAR of decision problems that can be solved in parallel polynomial time over the complex field collapses to P. As a result, one must first be able to show that there are VPSPACE families which are hard to evaluate in order to separate P from NP over C, or even from PAR.Comment: 14 page

The Multivariate Resultant is NP-hard in any Characteristic

The multivariate resultant is a fundamental tool of computational algebraic geometry. It can in particular be used to decide whether a system of n homogeneous equations in n variables is satisfiable (the resultant is a polynomial in the system's coefficients which vanishes if and only if the system is satisfiable). In this paper we present several NP-hardness results for testing whether a multivariate resultant vanishes, or equivalently for deciding whether a square system of homogeneous equations is satisfiable. Our main result is that testing the resultant for zero is NP-hard under deterministic reductions in any characteristic, for systems of low-degree polynomials with coefficients in the ground field (rather than in an extension). We also observe that in characteristic zero, this problem is in the Arthur-Merlin class AM if the generalized Riemann hypothesis holds true. In positive characteristic, the best upper bound remains PSPACE.Comment: 13 page

Difficulté du résultant et des grands déterminants

21 pagesLe résultant est un polynôme permettant de déterminer si plusieurs polynômes donnés ont une racine commune. Canny a pu donner un algorithme PSPACE calculant le résultant à l'aide de calculs de déterminants, mais pose la question de sa complexité exacte. On s'intéresse ici à donner une estimation plus fine de cette complexité. D'une part, on montre que le résultant est dans AM, et qu'il est NP-difficile sous réduction probabiliste. D'autre part, les matrices en jeu étant descriptibles par des circuits de taille raisonnable, on montre que le calcul du déterminant pour de telles matrices est PSPACE-complet

Représentations des polynômes, algorithmes et bornes inférieures

La complexité algorithmique est l'étude des ressources nécessaires le temps, la mémoire, pour résoudre un problème de manière algorithmique. Dans ce cadre, la théorie de la complexité algébrique est l'étude de la complexité algorithmique de problèmes de nature algébrique, concernant des polynômes.Dans cette thèse, nous étudions différents aspects de la complexité algébrique. D'une part, nous nous intéressons à l'expressivité des déterminants de matrices comme représentations des polynômes dans le modèle de complexité de Valiant. Nous montrons que les matrices symétriques ont la même expressivité que les matrices quelconques dès que la caractéristique du corps est différente de deux, mais que ce n'est plus le cas en caractéristique deux. Nous construisons également la représentation la plus compacte connue du permanent par un déterminant. D'autre part, nous étudions la complexité algorithmique de problèmes algébriques. Nous montrons que la détection de racines dans un système de n polynômes homogènes à n variables est NP-difficile. En lien avec la question VP = VNP ? , version algébrique de P = NP ? , nous obtenons une borne inférieure pour le calcul du permanent d'une matrice par un circuit arithmétique, et nous exhibons des liens unissant ce problème et celui du test d'identité polynomiale. Enfin nous fournissons des algorithmes efficaces pour la factorisation des polynômes lacunaires à deux variables.Computational complexity is the study of the resources time, memory, needed to algorithmically solve a problem. Within these settings, algebraic complexity theory is the study of the computational complexity of problems of algebraic nature, concerning polynomials. In this thesis, we study several aspects of algebraic complexity. On the one hand, we are interested in the expressiveness of the determinants of matrices as representations of polynomials in Valiant's model of complexity. We show that symmetric matrices have the same expressiveness as the ordinary matrices as soon as the characteristic of the underlying field in different from two, but that this is not the case anymore in characteristic two. We also build the smallest known representation of the permanent by a determinant.On the other hand, we study the computational complexity of algebraic problems. We show that the detection of roots in a system of n homogeneous polynomials in n variables in NP-hard. In line with the VP = VNP ? question, which is the algebraic version of P = NP? we obtain a lower bound for the computation of the permanent of a matrix by an arithmetic circuit, and we point out the links between this problem and the polynomial identity testing problem. Finally, we give efficient algorithms for the factorization of lacunary bivariate polynomials.LYON-ENS Sciences (693872304) / SudocSudocFranceF

Circuits versus Trees in Algebraic Complexity

. This survey is devoted to some aspects of the \P = NP ?" problem over the real numbers and more general algebraic structures. We argue that given a structure M , it is important to nd out whether NPM problems can be solved by polynomial depth computation trees, and if so whether these trees can be eciently simulated by circuits. Point location, a problem of computational geometry, comes into play in the study of these questions for several structures of interest. 1 Introduction In algebraic complexity one measures the complexity of an algorithm by the number of basic operations performed during a computation. The basic operations are usually arithmetic operations and comparisons, but sometimes transcendental functions are also allowed [21-23, 26]. Even when the set of basic operations has been xed, the complexity of a problem depends on the particular model of computation considered. The two main categories of interest for this paper are circuits and trees. In section 2 and..