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arXiv · 2010.10997

Rigid continuation paths II. Structured polynomial systems

Peter Bürgisser·Felipe Cucker·Pierre Lairez
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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.

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Peter Bürgisser, Felipe Cucker, Pierre Lairez. 2020-10-21. Rigid continuation paths II. Structured polynomial systems. https://doi.org/10.1017/fmp.2023.7

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