@article{BC-On-A-Problem-Posed-By-Steve-Smale,
Title = {On a Problem Posed by Steve Smale},
Author = {Peter Bürgisser and Felipe Cucker},
Pages = {1785-1836},
Year = {2011},
Journal = {Annals of Mathematics},
Volume = {174},
Number = {3},
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 Beltran 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, a la Beltran-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 $s^{-1}$, where $s$ 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.},
Url = {http://arxiv.org/abs/0909.2114},
Url2 = {http://www3.math.tu-berlin.de/algebra/work/smale17-final-corr.pdf}
}