Rigid continuation paths II. structured polynomial systems

Abstract

AbstractThis work studies the average complexity of solving structured polynomial systems that are characterised 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 complexityL. Combining both, we show that we can compute an approximate zero of a random structured polynomial system withnequations of degree at most${D}$innvariables with only$\operatorname {poly}(n, {D}) L$operations with high probability. This exceeds the expectations implicit in Smale’s 17th problem.