Smale’s 17th Problem: Average Polynomial Time to Compute Affine and Projective Solutions

In the series of papers [SS93a, SS93b, SS93c, SS94, SS96], Shub and Smale defined and studied in depth a homotopy method for solving systems of polynomial equations. Some articles preceding this new treatment were [Kan49, Sma86, Ren87, Kim89, Shu93]. Other authors have also treated this approach in [Mal94, Yak95, Ded97, BCSS98, Ded01, Ded06, MR02] and more recently in [Shu08, BS08]. In a previous paper [BP08], the authors furthered the program initiated in the series [SS93a] to [SS94], describing an algorithm that computes projective approximate zeros of systems of polynomial equations in polynomial running time, with bounded (small) probability of failure. In this paper, we give an updated version of some of the concepts introduced in [BP08], and we develop two important extensions of the results therein. On one hand, we describe a procedure that, instead of assuming a small probability of error, finds an approximate zero of systems of polynomial equations, on the average, in polynomial time (cf. Theorem 1.10). This improvement is made in order to answer explicitly the question posted by Smale in his 17th problem. On the other hand, we extend the result to the computation of affine solutions of systems of polynomial equations. This new result requires us to understand the probability distribution of the norm of the affine solutions of polynomial systems of equations (Theorem 1.9 below). These results may be summarized as follows.