Solving Equations Using Khovanskii Bases

TL;DR

This paper introduces a novel eigenvalue-based approach for solving structured polynomial equations on algebraic varieties using Khovanskii bases, extending existing methods for toric varieties and enhancing computer algebra techniques.

Contribution

It develops a new eigenvalue method leveraging Khovanskii bases for solving polynomial equations on varieties, generalizing algorithms for toric varieties and advancing computational algebra.

Findings

Effective eigenvalue method for polynomial equations on varieties

Extension of toric variety algorithms to broader classes

Applications in algebraic geometry and computer algebra

Abstract

We develop a new eigenvalue method for solving structured polynomial equations over any field. The equations are defined on a projective algebraic variety which admits a rational parameterization by a Khovanskii basis, e.g., a Grassmannian in its Pl\"ucker embedding. This generalizes established algorithms for toric varieties, and introduces the effective use of Khovanskii bases in computer algebra. We investigate regularity questions and discuss several applications.