Stable normal forms for polynomial system solving

arXiv:0812.0067v1 [cs.SC] 29 Nov 2008 Stable Normal Forms for Polynomial System Solving Bernard Mourrain Philippe Tr´ebuchet GALAAD, INRIA M´editerran´ee 2004 Route des Lucioles, BP 93, 06902 Sophia Antipolis, Cedex, France Bernard.Mourrain@inria.fr UPMC,LIP6, Equipe APR 4 place jussieu 75015 Paris, France Philippe.Trebuchet@lip6.fr July 23, 2008 Abstract This paper describes and analyzes a method for computing border bases of a zero-dimensional ideal I. The criterion used in the computation involves specific commutation polynomials and leads to an algorithm and an implementation extending the one provided in [29]. This general border basis algorithm weakens the monomial ordering requirement for Gr¨obner bases computa- tions. It is up to date the most general setting for representing quotient algebras, embedding into a single formalism Gr¨obner bases, Macaulay bases and new representation that do not fit into the previous categories. With this formalism we show how the syzygies of the border basis are generated by commutation relations. We also show that our construction of normal form is stable under small perturbations of the ideal, if the number of solutions remains constant. This new feature for a symbolic algorithm has a huge impact on the practical efficiency as it is illustrated by the experiments on classical benchmark polynomial systems, at the end of the paper. Keywords: Multivariate polynomial, quotient algebra, normal form, border basis, root-finding, symbolic-numeric computation. 1 Introduction Solving polynomial systems is the cornerstone of many applications in domains such as robotics, geometric modeling, signal processing, chromatology, structural molecular biology etc. In these prob- lems, the system has, most of the time, finitely many solutions and the equations often appear with approximate coefficients. From a computational point of view, it is an actual challenge to develop efficient and stable methods to solve such problems. First backward stability is expected. The computed solutions should be the exact solutions of a system in the neighborhood of the input system. Efficiency is also mandatory to tackle the encountered polynomial systems. One can expect for instance that the behavior of the method depends mainly on the number of solutions, and partially on other extrinsic parameters such as the number of variables. To handle the backward stability issue, one may consider classical numerical methods such as Newton-like iterations. However these local methods do not provide any guarantee of global conver- gence nor a complete description of all the roots. Algebraic methods, on the contrary, handle all the 1 roots simultaneously. They reduce the problem to computing the structure of the quotient algebra A of the polynomial ring modulo the ideal I generated by the input system [8]. Such a structure is given by a basis B of A as a vector space, and the tables of multiplication in A. Equivalently, it can be described by an algorithm of projection of the ring of polynomials K[x] onto the vector space ⟨B⟩ generated by B, along the ideal I. We call such a projection, a normal form for the ideal I. From the knowledge of the multiplication tables, we deduce either an exact encoding of the roots through a rational univariate representation [32], [12], or a numerical approximation by eigenvector computation, [1], [24], [34]. Since the eigencomputation can be considered as a numerically well- controlled process [13], the challenge becomes now to compute efficiently and in a stable way the quotient algebra structure A. In the family of algebraic methods, resultant-based techniques (see eg. [19], [9], [3]) exploit the properties of coefficients matrices of monomial multiples of the input equations in some specific degree. The table of multiplications are obtained by explicit Schur complements in these matrices [9]. Their construction is deeply linked to the geometry of the underlying variety that make these methods very tolerant against small perturbations. Unfortunately they heavily rely on genericity hypotheses that reduce their applicability. Moreover, the size of the constructed matrices usually grows exponentially with the number of variables. To avoid such a pitfall, so-called H-bases have also been studied [20], [22]. They proceed degree by degree and stop when the terms of highest degree of the computed polynomials generate the terms of highest degree of all the ideal I. The stopping criterion requires the computation of generators of the syzygies of these highest degree terms. Though it also yields a basis of the quotient ring A, without any apriori knowledge on its dimension, practically speaking, it also suffers from the swelling of the size of the linear systems to be solved. The approach can be refined further by using a grading for which the highest term of a polynomial is a monomial. This leads to Gr¨obner basis computation (see eg. [6]). This approach also yields a basis of the quotient algebra A an

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