Title: An optimal numerical algorithm for solving polynomial systems
Abstract: Numerical homotopy continuation is a useful numerical algorithm for computing the solutions of a system of polynomial equations. Such solution sets are known as varieties. Homotopies compute varieties by tracking paths from the solutions of a similar, pre-solved system. Generally, homotopies may track extraneous paths, which wastes computational resources. A homotopy is optimal if paths are smooth and there are no extraneous paths. Embedded toric degenerations are one source for optimal homotopy algorithms. In particular, if a variety has a toric degeneration, then there is an optimal homotopy for computing linear sections of that variety. There is a toric degeneration for any variety which has an associated finite Khovanskii basis. This work provides the appropriate embeddings for the Khovanskii toric degeneration and gives the corresponding optimal homotopy algorithm for computing a linear section of the variety. This is joint work with Michael Burr (Clemson University) and Frank Sottile (Texas A&M University).