Smale's Fundamental Theorem of Algebra reconsidered
  In his 1981 Fundamental Theorem of Algebra paper Steve Smale initiated the complexity theory of finding a solution of polynomial equations of one complex variable by a variant of Newton's method. In this paper we reconsider his algorithm in the light of work done in the intervening years. Smale's upper bound estimate was infinite average cost. Our's is polynomial in the B\'ezout number and the dimension of the input. Hence polynomial for any range of dimensions where the B\'ezout number is polynomial in the input size. In particular not just for the case that Smale considered but for a range of dimensions as considered by B\"urgisser-Cucker where the max of the degrees is greater than or equal to $n^{1+\epsilon}$ for some fixed $\epsilon$. It is possible that Smale's algorithm is polynomial cost in all dimensions and our main theorem raises some problems that might lead to a proof of such a theorem. 
Diego Armentano, Michael Shub
math.NA math.AG math.CV math.DS