Numerical homotopies from Khovanskii bases

TL;DR

This paper introduces numerical homotopy algorithms leveraging Khovanskii bases and Anderson's degeneration to efficiently solve polynomial systems on varieties, with implementations demonstrated in Macaulay2.

Contribution

It develops novel homotopy continuation algorithms based on Khovanskii bases and Anderson's degeneration, extending toric homotopy methods to broader classes of varieties.

Findings

Algorithms successfully solve polynomial systems on varieties with Khovanskii bases.

Implementation in Macaulay2 demonstrates practical applicability.

Extensions to weighted projective spaces are achieved.

Abstract

We present numerical homotopy continuation algorithms for solving systems of equations on a variety in the presence of a finite Khovanskii basis. These 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.