Smale's Fundamental Theorem of Algebra reconsidered

TL;DR

This paper reevaluates Smale's algorithm for solving polynomial equations, demonstrating it has polynomial complexity under broader conditions and raising questions about its universal polynomial cost across all dimensions.

Contribution

The paper provides a new complexity analysis of Smale's algorithm, showing it is polynomial in the Bézout number and input dimension, extending previous bounds.

Findings

Smale's upper bound estimate was infinite average cost.

The new analysis shows polynomial complexity in the Bézout number and dimension.

Raises open problems about polynomial cost of Smale's algorithm in all dimensions.

Abstract

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.