Complexity Of Computing Complex Zeros Of Structured Polynomial Systems

          @misc{ scivideos_18808,
            doi = {},
            url = {https://simons.berkeley.edu/talks/complexity-computing-complex-zeros-structured-polynomial-systems},
            author = {},
            keywords = {},
            language = {en},
            title = {Complexity Of Computing Complex Zeros Of Structured Polynomial Systems},
            publisher = {The Simons Institute for the Theory of Computing},
            year = {2021},
            month = {dec},
            note = {18808 see, \url{https://scivideos.org/index.php/Simons-Institute/18808}}
          }
          

Abstract

We study 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 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 d in n variables with only poly(n,d) L operations with high probability. This exceeds the expectations implicit in Smale's 17th problem. This is joint work with Felipe Cucker and Pierre Lairez, see arXiv:2010.10997.