Rigid continuation paths II. Structured polynomial systems
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.
Cite
@article{arxiv.2010.10997,
title = {Rigid continuation paths II. Structured polynomial systems},
author = {Peter Bürgisser and Felipe Cucker and Pierre Lairez},
journal= {arXiv preprint arXiv:2010.10997},
year = {2023}
}