  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.
