Smale's 17th problem

2012

These pages contain a short overview on the state of the art of efficient numerical analysis methods that solve systems of multivariate polynomial equations. We focus on the work of Steve Smale who initiated this research framework, and on the collaboration between Stephen Smale and Michael Shub, which set the foundations of this approach to polynomial system--solving, culminating in the more recent advances of Carlos Beltran, Luis Miguel Pardo, Peter Buergisser and Felipe Cucker.

SIAM Journal on Computing, 2003

We propose new Las Vegas randomized algorithms for the solution of a square nondegenerate system of equations, with well-separated roots. The algorithms use O(δ 3 n D 2 log(D) log(b)) arithmetic operations (in addition to the operations required to compute the normal form of the boundary monomials modulo the ideal) to approximate all real roots of the system as well as all roots lying in a fixed n-dimensional box or disc. Here D is an upper bound on the number of all complex roots of the system (e.g., Bezout or Bernshtein bound), δ is the number of real roots or the roots lying in the box or disc, and = 2 −b is the required upper bound on the output errors. For computing the normal form modulo the ideal, the efficient practical algorithms of [B. Mourrain and P. Trébuchet, in 61-88] can be applied. We also yield the bound O(3 n D 2 log(D)) on the complexity of counting the numbers of all roots in a fixed box (disc) and all real roots. For a large class of inputs and typically in practical computations, the factor δ is much smaller than D, δ = o(D). This improves by the order of magnitude the known complexity estimates of the order of at least 3 n D 4 + D 3 log(b) or D 4 , which so far are the record estimates even for the approximation of a single root of a system and for each of the cited counting problems, respectively. Our progress relies on proposing several novel techniques. In particular, we exploit the structure of matrices associated to a given polynomial system and relate it to the associated linear operators, dual space of linear forms, and normal forms of polynomials in the quotient algebra; furthermore, our techniques support the new nontrivial extension of the matrix sign and quadratic inverse power iterations to the case of multivariate polynomial systems, where we emulate the recursive splitting of a univariate polynomial into factors of smaller degree.

Foundations of Computational Mathematics, 2010

In the forthcoming paper of Beltrán and Pardo, the average complexity of linear homotopy methods to solve polynomial equations with random initial input (in a sense to be described below) was proven to be finite, and even polynomial in the size of the input. In this paper, we prove that some other higher moments are also finite. In particular, we show that the variance is polynomial in the size of the input.

We describe an Average Las Vegas algorithm to compute several zeros of polynomial systems, and analyze its complexity and Shannon Entropy. We also derive the first universal algorithm to solve this kind of systems in average running time linear in the Bézout number and polynomial in the input length for dense encoding of polynomials.

Journal of Complexity, 2003

A new technique for the geometry of numbers is exhibited. This technique provides sharp estimates on the number of bounded height rational points in subsets of projective space whose ''projective cone'' is semi-algebraic. This technique improves existing techniques as the one introduced by Davenport in (J. London Math. Soc. 26 (1951) 179). As main outcome, we conclude that systems of rational polynomial equations of bounded bit length have polynomial size approximate zeros on the average. We also conclude that the average number of projective real solutions of systems of rational polynomial equations of bounded bit length equals the square root of the Be´zout number of the given system. r

Your article is protected by copyright and all rights are held exclusively by SFoCM. This eoffprint is for personal use only and shall not be self-archived in electronic repositories. If you wish to self-archive your article, please use the accepted manuscript version for posting on your own website. You may further deposit the accepted manuscript version in any repository, provided it is only made publicly available 12 months after official publication or later and provided acknowledgement is given to the original source of publication and a link is inserted to the published article on Springer's website. The link must be accompanied by the following text: "The final publication is available at link.springer.com". Abstract Smale's 17th problem asks for an algorithm which finds an approximate zero of polynomial systems in average polynomial time (see Smale in Mathematical problems for

Foundations of Computational Mathematics, 2011

We prove a new complexity bound, polynomial on the average, for the problem of finding an approximate zero of systems of polynomial equations. The average number of Newton steps required by this method is almost linear in the size of the input. We show that the method can also be used to approximate several or all the solutions of non-degenerate systems, and prove that this last task can be done in running time which is linear in the Bézout number of the system, on the average.

Foundations of Computational Mathematics, 2003

Elimination theory was at the origin of algebraic geometry in the nineteenth century and now deals with the algorithmic solving of multivariate polynomial equation systems over the complex numbers or, more generally, over an arbitrary algebraically closed field. In this paper we investigate the intrinsic sequential time complexity of universal elimination procedures for arbitrary continuous data structures encoding input and output objects of elimination theory (i.e., polynomial equation systems) and admitting the representation of certain limit objects. Our main result is the following: let there be given such a data structure and together with this data structure a universal elimination algorithm, say P, solving arbitrary parametric polynomial equation systems. Suppose that the algorithm P avoids “unnecessary” branchings and that P admits the efficient computation of certain natural limit objects (as, e.g., the Zariski closure of a given constructible algebraic set or the parametric greatest common divisor of two given algebraic families of univariate polynomials). Then P$ cannot be a polynomial time algorithm. The paper contains different variants of this result and discusses their practical implications.

Proceedings of the thirtieth annual ACM symposium on Theory of computing - STOC '98, 1998

We propose new Las Vegas randomized algorithms for the solution of a multivariate generic or sparse polynomial system of equations. The algorithms use O (( +4 n )3 n D 2 log b) arithmetic operations to approximate all real roots of the system as well as all roots lying in a fixed n-dimensional box or disc. Here D is an upper bound on the number of all the roots of the system, is the number of real roots or the roots lying in the box or disc, = 2 ?b is the required upper bound on the output errors, and O (s) stands for O(s log c s), c being a constant independent of s. We also yield the bounds O (12 n D 2 ) for the complexity of counting the numbers of all roots in a fixed box (disc) and all real roots and O (12 n D 2 log b)

Contemporary Mathematics, 2011

These pages contain a short overview on the state of the art of efficient Numerical Analysis methods that solve systems of multivariate polynomial equations. We focus on the work of Steve Smale who initiated this research framework, and on the collaboration between Steve Smale and Mike Shub, which set the foundations of this approach to polynomial system-solving.