Rigid continuation paths II. Structured polynomial systems

TL;DR

This paper develops a continuation algorithm for solving structured polynomial systems efficiently, showing that approximate zeros can be computed with polynomial complexity relative to system size and evaluation cost, surpassing prior expectations.

Contribution

It introduces a new algorithm and a universal model for random structured polynomial systems, demonstrating polynomial-time solutions based on evaluation complexity.

Findings

Efficient continuation algorithm for structured polynomial systems

Universal model for random polynomial systems with prescribed evaluation complexity

Polynomial-time computation of approximate zeros with high probability

Abstract

This work studies the average complexity of solving structured polynomial systems that are characterized by a low evaluation cost, as opposed to the dense random model previously used. Firstly, we design a continuation algorithm that computes, with high probability, an approximate zero of a polynomial system given only as black-box evaluation program. Secondly, we introduce a universal model of random polynomial systems with prescribed evaluation complexity L. Combining both, we show that we can compute an approximate zero of a random structured polynomial system with n equations of degree at most {\delta} in n variables with only poly(n, {\delta}) L operations with high probability. This exceeds the expectations implicit in Smale's 17th problem.