Smale's Fundamental Theorem of Algebra reconsidered

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 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 $n^{1+\epsilon }$ 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.

This paper has been withdrawn by the authors because Lemma 11, for p>1, is incorrect

References in corpus (2)

• Complexity of sparse polynomial solving: homotopy on toric varieties and the condition metric
• Adaptative Step Size Selection for Homotopy Methods to Solve Polynomial Equations

Cited by in corpus (1)

• Complexity of Path-Following Methods for the Eigenvalue Problem