Fast Computation of Zeros of Polynomial Systems with Bounded Degree under Finite-precision

TL;DR

This paper extends an algorithm for finding approximate zeros of bounded-degree polynomial systems to finite-precision arithmetic, demonstrating it maintains polynomial-time performance with manageable precision requirements.

Contribution

It introduces a finite-precision adaptation of a polynomial-time algorithm for polynomial zeros, ensuring practical applicability with bounded computational resources.

Findings

Algorithm operates within same time bounds as infinite-precision version

Average required precision is polynomial in input size

Maintains accuracy and efficiency in finite-precision arithmetic

Abstract

A solution for Smale's 17th problem, for the case of systems with bounded degree was recently given. This solution, an algorithm computing approximate zeros of complex polynomial systems in average polynomial time, assumed infinite precision. In this paper we describe a finite-precision version of this algorithm. Our main result shows that this version works within the same time bounds and requires a precision which, on the average, amounts to a polynomial amount of bits in the mantissa of the intervening floating-point numbers.