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 linear homotopy algorithm, call it LV, doing so. Recently, we extended this result in several directions, making systematic use of the nice properties of Gaussian probability distributions.

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 1/sigma, where sigma controls the size of of the random perturbation of the input systems. Secondly, we perform a condition-based analysis of LV. That is, we give a bound on the expected running time of LV on input f that, besides on the dimension parameters, only depends on the condition of the input system f.

Finally, 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 (joint work with Felipe Cucker, CityU Hong Kong).