Fall 2021

Complexity Of Computing Complex Zeros Of Structured Polynomial Systems

Wednesday, Dec. 1, 2021 9:30 am10:30 am

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Peter Buergisser (TU Berlin)


Calvin Lab Auditorium

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.