Solving polynomial equations in smoothed polynomial time and a near solution to Smale's 17th problem

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, call it LV, doing so. In this paper we further extend this result in several directions. Firstly, 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 the random perturbation of the input systems. Secondly, 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. Thirdly, 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. © 2010 ACM.