On the complexity of computing with zero-dimensional triangular sets

We study the complexity of some fundamental operations for triangular sets in dimension zero. Using Las-Vegas algorithms, we prove that one can perform such operations as change of order, equiprojectable decomposition, or quasi-inverse computation with a cost that is essentially that of modular composition. Over an abstract field, this leads to a subquadratic cost (with respect to the degree of the underlying algebraic set). Over a finite field, in a boolean RAM model, we obtain a quasi-linear running time using Kedlaya and Umans' algorithm for modular composition. Conversely, we also show how to reduce the problem of modular composition to change of order for triangular sets, so that all these problems are essentially equivalent. Our algorithms are implemented in Maple; we present some experimental results

On the Complexity of Solving Zero-Dimensional Polynomial Systems via Projection

Given a zero-dimensional polynomial system consisting of n integer polynomials in n variables, we propose a certified and complete method to compute all complex solutions of the system as well as a corresponding separating linear form l with coefficients of small bit size. For computing l, we need to project the solutions into one dimension along O(n) distinct directions but no further algebraic manipulations. The solutions are then directly reconstructed from the considered projections. The first step is deterministic, whereas the second step uses randomization, thus being Las-Vegas. The theoretical analysis of our approach shows that the overall cost for the two problems considered above is dominated by the cost of carrying out the projections. We also give bounds on the bit complexity of our algorithms that are exclusively stated in terms of the number of variables, the total degree and the bitsize of the input polynomials

Certified Algorithms for proving the structural stability of two dimensional systems possibly with parameters

International audienceIn [1], a new method for testing the structural stability of multidimensional systems has been presented. The key idea of this method is to reduce the problem of testing the structural stability to that of deciding if an algebraic set has real points. Following the same idea, we consider in this work the specific case of two-dimensional systems and focus on the practical efficiency aspect. For such systems, the problem of testing the stability is reduced to that of deciding if a bivariate algebraic system with finitely many solutions has real ones. Our first contribution is an algorithm that answers this question while achieving practical efficiency. Our second contribution concerns the stability of two dimensional systems with parameters. More precisely, given a two-dimensional system depending on a set of parameters, we present a new algorithm that computes regions of the parameter space in which the considered system is structurally stable

A Generic Position Based Method for Real Root Isolation of Zero-Dimensional Polynomial Systems

We improve the local generic position method for isolating the real roots of a zero-dimensional bivariate polynomial system with two polynomials and extend the method to general zero-dimensional polynomial systems. The method mainly involves resultant computation and real root isolation of univariate polynomial equations. The roots of the system have a linear univariate representation. The complexity of the method is $\tilde{O}_B(N^{10})$ for the bivariate case, where $N=\max(d,\tau)$, $d$ resp., $\tau$ is an upper bound on the degree, resp., the maximal coefficient bitsize of the input polynomials. The algorithm is certified with probability 1 in the multivariate case. The implementation shows that the method is efficient, especially for bivariate polynomial systems.Comment: 24 pages, 5 figure

On the Complexity of Solving Zero-Dimensional Polynomial Systems via Projection

Given a zero-dimensional polynomial system consisting of n integer polynomials in n variables, we propose a certified and complete method to compute all complex solutions of the system as well as a corresponding separating linear form l with coefficients of small bit size. For computing l, we need to project the solutions into one dimension along O(n) distinct directions but no further algebraic manipulations. The solutions are then directly reconstructed from the considered projections. The first step is deterministic, whereas the second step uses randomization, thus being Las-Vegas. The theoretical analysis of our approach shows that the overall cost for the two problems considered above is dominated by the cost of carrying out the projections. We also give bounds on the bit complexity of our algorithms that are exclusively stated in terms of the number of variables, the total degree and the bitsize of the input polynomials

Computing Limit Points of Quasi-components of Regular Chains and its Applications

Computing limit is a fundamental task in mathematics and different mathematical concepts are defined in terms of limit computations. Among these mathematical concepts, we are interested in three different types of limit computations: first, computing the limit points of solutions of polynomial systems represented by regular chains, second, computing tangent cones of space curves at their singular points which can be viewed as computing limit of secant lines, and third, computing the limit of real multivariate rational functions. For computing the limit of solutions of polynomial systems represented by regular chains, we present two different methods based on Puiseux series expansions and linear changes of coordinates. The first method, which is based on Puiseux series expansions, addresses the problem of computing real and complex limit points corresponding to regular chains of dimension one. The second method studies regular chains under changes of coordinates. It especially computes the limit points corresponding to regular chains of dimension higher than one for some cases. we consider strategies where these changes of coordinates can be either generic or guided by the input. For computing the Puiseux parametrizations corresponding to regular chains of dimension one, we rely on extended Hensel construction (EHC). The Extended Hensel Construction is a procedure which, for an input bivariate polynomial with complex coefficients, can serve the same purpose as the Newton-Puiseux algorithm, and, for the multivariate case, can be seen as an effective variant of Jung-Abhyankar Theorem. We show that the EHC requires only linear algebra and univariate polynomial arithmetic. We deduce complexity estimates and report on a software implementation together with experimental results. We also outline a method for computing the tangent cone of a space curve at any of its points. We rely on the theory of regular chains and Puiseux series expansions. Our approach is novel in that it explicitly constructs the tangent cone at arbitrary and possibly irrational points without using a Standard basis. We also present an algorithm for determining the existence of the limit of a real multivariate rational function q at a given point which is an isolated zero of the denominator of q. When the limit exists, the algorithm computes it, without making any assumption on the number of variables. A process, which extends the work of Cadavid, Molina and V´elez, reduces the multivariate setting to computing limits of bivariate rational functions. By using regular chain theory and triangular decomposition of semi-algebraic systems, we avoid the computation of singular loci and the decomposition of algebraic sets into irreducible components

Bivariate systems and topology of plane curves: algebraic and numerical methods

The work presented in this thesis belongs to the domain of non-linear computational geometry in lowdimension. More precisely it focuses on solving bivariate systems and computing the topology of curvesin the plane. When the input is given by polynomials, the natural tools come from computer algebra.Our contributions are algorithms proven efficient in a deterministic or a Las Vegas settings together witha practical efficient software for topology certified drawing of a plane algebraic curve. When the input isnot restricted to be polynomials but given by interval functions, we design algorithms based on certifiednumerical approches using subdivision and interval arithmetic. The input is then required to fulfill somegeneric assumptions and our algorithms are certified in the sense that they terminate if and only if theassumptions are satisfied.Le travail présenté dans cette thèse appartient au domaine de la géométrie computationnelle non linéaireen petite dimension. Plus précisément, il se concentre sur la résolution de systèmes bivariés et le calcul dela topologie des courbes dans le plan. Lorsque l’entrée est donnée par des polynômes, les outils naturelsproviennent du calcul formel. Nos contributions sont des algorithmes dont l’efficacité a été prouvée dansun cadre déterministe ou Las Vegas, ainsi qu’un logiciel efficace pour le dessin certifié de la topologied’une courbe algébrique plane. Lorsque les données d’entrée ne sont pas limitées aux polynômes maissont données par des fonctions d’intervalles, nous concevons des algorithmes basés sur des approchesnumériques certifiées utilisant la subdivision et l’arithmétique d’intervalles. L’entrée doit alors satisfairecertaines hypothèses génériques et nos algorithmes sont certifiés dans le sens où ils se terminent si etseulement si les hypothèses sont satisfaites