Rigid continuation paths II. Structured polynomial systems [article]

Peter Bürgisser, Felipe Cucker, Pierre Lairez
2021 arXiv   pre-print
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
more » ... compute an approximate zero of a random structured polynomial system with n equations of degree at most δ in n variables with only poly(n, δ) L operations with high probability. This exceeds the expectations implicit in Smale's 17th problem.
arXiv:2010.10997v2 fatcat:k2oucawmqfdrlkp2kduwlgofdu