Numerical homotopies from Khovanskii bases

Abstract

We present numerical homotopy continuation algorithms for solving systems of equations on a variety in the presence of a finite Khovanskii basis. These homotopies take advantage of Anderson’s flat degeneration to a toric variety. When Anderson’s degeneration embeds into projective space, our algorithm is a special case of a general toric two-step homotopy algorithm. When Anderson’s degeneration is embedded in a weighted projective space, we explain how to lift to a projective space and construct an appropriate modification of the toric homotopy. Our algorithms are illustrated on several examples using Macaulay2.