Smale's Fundamental Theorem of Algebra reconsidered
For researchers in complexity theory and numerical analysis, this provides a polynomial average-case complexity result for a classic algorithm, extending previous work and suggesting potential for full polynomiality.
This paper revisits Smale's 1981 algorithm for solving polynomial equations via Newton's method, improving the average cost bound from infinite to polynomial in the Bézout number and input dimension, specifically for dimensions where the maximum degree is at least n^{1+ε}.
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 Bézout number and the dimension of the input. Hence polynomial for any range of dimensions where the Bézout 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 Bürgisser-Cucker where the max of the degrees is greater than or equal to $n^{1+ε}$ for some fixed $ε$. 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.