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 Plücker 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.

Given a tropical line $L$ and a smooth tropical surface $X$, we look at the position of $L$ on $X$. We introduce its primal and dual motifs which are respectively a decorated graph and a subcomplex of the dual triangulation of $X$. They encode the combinatorial position of $L$ on $X$. We classify all possible motifs of tropical lines on general smooth tropical surfaces. This classification allows us to give an upper bound for the number of tropical lines on a general smooth tropical surface with a given subdivision. We focus in particular on surfaces of degree three. As a concrete example, we look at tropical cubic surfaces dual to a fixed honeycomb triangulation, showing that a general surface contains exactly $27$ tropical lines.

Abstract Smooth tropical cubic surfaces are parametrized by maximal cones in the unimodular secondary fan of the triple tetrahedron. There are $$344\, 843 \,867$$ 344 843 867 such cones, organized into a database of $$14\,373\,645$$ 14 373 645 symmetry classes. The Schläfli fan gives a further refinement of these cones. It reveals all possible patterns of lines on tropical cubic surfaces, thus serving as a combinatorial base space for the universal Fano variety. This article develops the relevant theory and offers a blueprint for the analysis of big data in tropical geometry.