On a problem posed by Steve Smale

Abstract

Abstract. The 17th of the problems proposed by Steve Smale for the 21st century asks for the existence of a deterministic algorithm computing an approximate solution of a system of n complex polynomials in n unknowns in time polynomial, on the average, in the size N of the input system. A partial solution to this problem was given by Carlos Beltrán and Luis Miguel Pardo who exhibited a randomized algorithm doing so. In this paper we further extend this result in several directions. Firstly, we exhibit a linear homotopy algorithm that efficiently implements a non-constructive idea of Mike Shub. This algorithm is then used in a randomized algorithm, call it LV,à la Beltrán-Pardo. Secondly, we perform a smoothed analysis (in the sense of Spielman and Teng) of algorithm LV and prove that its smoothed complexity is polynomial in the input size and σ −1 , where σ controls the size of of the random perturbation of the input systems. Thirdly, we perform a condition-based analysis of LV. That is, we give a bound, for each system f , of the expected running time of LV with input f . In addition to its dependence on N this bound also depends on the condition of f . Fourthly, and to conclude, we return to Smale's 17th problem as originally formulated for deterministic algorithms. We exhibit such an algorithm and show that its average complexity is N O(log log N) . This is nearly a solution to Smale's 17th problem.

Stochastic perturbations and smooth condition numbers

“…The expected value of κ 2 norm (f ) is an essential ingredient in the complexity analysis of pathfollowing methods (cf. Shub-Smale [14], Beltran-Pardo [3], and recently Bürgisser-Cucker [6]). In Beltran-Pardo [3], the authors proved that…”

On the zeta Mahler measure function of the Jacobian determinant, condition numbers and the height of the generic discriminant

“…A reader interested in more detailed arguments supporting the use of BombieriWeyl Hermitian product, and the probability distributions induced either on S(H (d) ) or S (d) , may follow [6,7,9,13,14,23,55,58] and references therein. Now we introduce several functions defined either in H (d) , S(H (d) ) or S (d) , through functions that depend on the set of projective zeros V P ( f ) associated to f .…”

“…[60]). Among the many contributions dealing with this problem we may cite [5][6][7]13,59] and the books [9,14] and references therein. The following zeta Mahler measure function was clued in [7].…”

Random sampling in computational algebra: Helly numbers and violator spaces

“…Smoothed analysis combines the worst-case and average-case algorithmic analyses by measuring the expected performance of algorithms under slight random perturbations of worst-case inputs. Of course, probabilistic analysis, and smoothed analysis in particular, has been used in computational algebraic geometry for some time now, see e.g., the elegant work in [8,9,15]. The aim of this paper is to import a randomized sampling framework from geometric optimization to applied computational algebra, and demonstrate its usefulness on two problems.…”