8 research outputs found
Factoring bivariate lacunary polynomials without heights
We present an algorithm which computes the multilinear factors of bivariate
lacunary polynomials. It is based on a new Gap Theorem which allows to test
whether a polynomial of the form P(X,X+1) is identically zero in time
polynomial in the number of terms of P(X,Y). The algorithm we obtain is more
elementary than the one by Kaltofen and Koiran (ISSAC'05) since it relies on
the valuation of polynomials of the previous form instead of the height of the
coefficients. As a result, it can be used to find some linear factors of
bivariate lacunary polynomials over a field of large finite characteristic in
probabilistic polynomial time.Comment: 25 pages, 1 appendi
Lacunaryx: Computing bounded-degree factors of lacunary polynomials
In this paper, we report on an implementation in the free software Mathemagix
of lacunary factorization algorithms, distributed as a library called
Lacunaryx. These algorithms take as input a polynomial in sparse
representation, that is as a list of nonzero monomials, and an integer , and
compute its irreducible degree- factors. The complexity of these
algorithms is polynomial in the sparse size of the input polynomial and .Comment: 6 page
Computing low-degree factors of lacunary polynomials: a Newton-Puiseux approach
We present a new algorithm for the computation of the irreducible factors of
degree at most , with multiplicity, of multivariate lacunary polynomials
over fields of characteristic zero. The algorithm reduces this computation to
the computation of irreducible factors of degree at most of univariate
lacunary polynomials and to the factorization of low-degree multivariate
polynomials. The reduction runs in time polynomial in the size of the input
polynomial and in . As a result, we obtain a new polynomial-time algorithm
for the computation of low-degree factors, with multiplicity, of multivariate
lacunary polynomials over number fields, but our method also gives partial
results for other fields, such as the fields of -adic numbers or for
absolute or approximate factorization for instance.
The core of our reduction uses the Newton polygon of the input polynomial,
and its validity is based on the Newton-Puiseux expansion of roots of bivariate
polynomials. In particular, we bound the valuation of where is
a lacunary polynomial and a Puiseux series whose vanishing polynomial
has low degree.Comment: 22 page
Quantitative Aspects of Sums of Squares and Sparse Polynomial Systems
Computational algebraic geometry is the study of roots of polynomials and polynomial systems. We are familiar with the notion of degree, but there are other ways to consider a polynomial: How many variables does it have? How many terms does it have? Considering the sparsity of a polynomial means we pay special attention to the number of terms. One can sometimes profit greatly by making use of sparsity when doing computations by utilizing tools from linear programming and integer matrix factorization. This thesis investigates several problems from the point of view of sparsity. Consider a system F of n polynomials over n variables, with a total of n + k distinct exponent vectors over any local field L. We discuss conjecturally tight bounds on the maximal number of non-degenerate roots F can have over L, with all coordinates having fixed phase, as a function of n, k, and L only. In particular, we give new explicit systems with number of roots approaching the best known upper bounds. We also give a complete classification for when an n-variate n + 2-nomial positive polynomial can be written as a sum of squares of polynomials. Finally, we investigate the problem of approximating roots of polynomials from the viewpoint of sparsity by developing a method of approximating roots for binomial systems that runs more efficiently than other current methods. These results serve as building blocks for proving results for less sparse polynomial systems
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