A deterministic algorithm to compute approximate roots of polynomial systems in polynomial average time

TL;DR

This paper presents a deterministic algorithm that efficiently computes approximate roots of polynomial systems in average polynomial time, providing a constructive solution to Smale's 17th problem.

Contribution

It derandomizes an existing algorithm, leveraging input randomness to achieve polynomial average time complexity in the Blum-Shub-Smale model.

Findings

Provides a deterministic algorithm for polynomial root approximation

Achieves average polynomial time complexity

Offers a constructive solution to Smale's 17th problem

Abstract

We describe a deterministic algorithm that computes an approximate root of n complex polynomial equations in n unknowns in average polynomial time with respect to the size of the input, in the Blum-Shub-Smale model with square root. It rests upon a derandomization of an algorithm of Beltr\'an and Pardo and gives a deterministic affirmative answer to Smale's 17th problem. The main idea is to make use of the randomness contained in the input itself.