Condition Numbers for the Cube. I: Univariate Polynomials and Hypersurfaces

TL;DR

This paper introduces a new condition number framework based on the cube norm for analyzing the complexity of solving univariate polynomials and hypersurfaces, demonstrating polynomial average-case bounds for specific algorithms.

Contribution

It develops a novel condition-based complexity framework applicable to sparse and Gaussian polynomials, extending the analysis tools in numerical algebraic geometry.

Findings

Polynomial average runtime for Plantinga-Vegter algorithm on random polynomials

Polynomial bounds on subdivision tree size for Descartes' solver

Bounds hold for all higher moments, not just average case

Abstract

The condition-based complexity analysis framework is one of the gems of modern numerical algebraic geometry and theoretical computer science. Among the challenges that it poses is to expand the currently limited range of random polynomials that we can handle. Despite important recent progress, the available tools cannot handle random sparse polynomials and Gaussian polynomials, that is polynomials whose coefficients are i.i.d. Gaussian random variables. We initiate a condition-based complexity framework based on the norm of the cube that is a step in this direction. We present this framework for real hypersurfaces and univariate polynomials. We demonstrate its capabilities in two problems, under very mild probabilistic assumptions. On the one hand, we show that the average run-time of the Plantinga-Vegter algorithm is polynomial in the degree for random sparse (alas a restricted sparseness structure) polynomials and random Gaussian polynomials. On the other hand, we study the size of the subdivision tree for Descartes' solver and run-time of the solver by Jindal and Sagraloff (arXiv:1704.06979). In both cases, we provide a bound that is polynomial in the size of the input (size of the support plus the logarithm of the degree) not only for the average but also for all higher moments.