On a Problem Posed by Steve Smale

Foundations of Computational Mathematics, 2008

Smale's 17th Problem asks “Can a zero of n complex polynomial equations in n unknowns be found approximately, on the average [for a suitable probability measure on the space of inputs], in polynomial time with a uniform algorithm?” We present a uniform probabilistic algorithm for this problem and prove that its complexity is polynomial. We thus obtain a partial positive solution to Smale's 17th Problem.

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Journal of The American Mathematical Society, 2009

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.

1987

Chemical Engineering Science, 1973

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.

Chinese Annals of Mathematics, Series B, 2007

A linear system arising from a polynomial problem in the approximation theory is studied, and the necessary and sufficient conditions for existence and uniqueness of its solutions are presented. Together with a class of determinant identities, the resulting theory is used to determine the unique solution to the polynomial problem. Some homogeneous polynomial identities as well as results on the structure of related polynomial ideals are just by-products.

European Journal of Operational Research, 2005

We present a short overview on polynomial approximation of NP-hard problems. We present the main approximability classes together with examples of problems belonging to them. We also describe the general concept of approximability preserving reductions together with a discussion about their capacities and their limits. Finally, we present a quick description of what it is commonly called inapproximability results. Such results provide limits on the approximability of the problems tackled.

2011

This paper introduces the notions of approximate and optimal approximate zero polynomial of a polynomial matrix by deploying recent results on the approximate GCD of a set of polynomials [1] and the exterior algebra [4] representation of polynomial matrices. The results provide a new definition for the "approximate", or "almost" zeros of polynomial matrices and provide the means for computing the distance from non-coprimeness of a polynomial matrix. The computational framework is expressed as a distance problem in a projective space. The general framework defined for polynomial matrices provides a new characterization of approximate zeros and decoupling zeros [2], [4] of linear systems and a process leading to computation of their optimal versions. The use of restriction pencils provides the means for defining the distance of state feedback (output injection) orbits from uncontrollable (unobservable) families of systems, as well as the invariant versions of the "approximate decoupling polynomials".