Robust certified numerical homotopy tracking

TL;DR

This paper introduces a fully rigorous homotopy algorithm for solving polynomial systems, ensuring approximate zeros with rational or integer coordinates and analyzing its complexity in terms of path condition and input size.

Contribution

It presents the first completely rigorous, certified homotopy tracking algorithm with proven complexity bounds in the Turing machine model.

Findings

Algorithm guarantees approximate zeros with rational or integer coordinates.

Total bit complexity is linear in path length and polynomial in condition number and input size.

Provides a rigorous foundation for numerical homotopy methods in polynomial solving.

Abstract

We describe, for the first time, a completely rigorous homotopy (path--following) algorithm (in the Turing machine model) to find approximate zeros of systems of polynomial equations. If the coordinates of the input systems and the initial zero are rational our algorithm involves only rational computations and if the homotopy is well posed an approximate zero with integer coordinates of the target system is obtained. The total bit complexity is linear in the length of the path in the condition metric, and polynomial in the logarithm of the maximum of the condition number along the path, and in the size of the input.