Solving equations using Khovanskii bases

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 & uuml;cker 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. (c) 2024 The Author(s). Published by Elsevier Ltd. This is an open access article under the CC BY license (http:// creativecommons .org /licenses /by /4 .0/).