Determining Tropical Hypersurfaces

TL;DR

This paper investigates when points in tropical affine space uniquely determine a tropical hypersurface, introducing multiplicity, and providing combinatorial and algebraic criteria for uniqueness and non-singularity.

Contribution

It introduces a multiplicity concept for points, links hypersurface determination to dual complex connectivity, and develops tropical linear algebra with a Cramer's rule analogue.

Findings

Connectedness of a sub-complex indicates uniqueness of the hypersurface.

A notion of non-singularity for tropical matrices is established.

A tropical Cramer's rule theorem is proved.

Abstract

We consider the question of when points in tropical affine space uniquely determine a tropical hypersurface. We introduce a notion of multiplicity of points so that this question may be meaningful even if some of the points coincide. We give a geometric/combinatorial way and a tropical linear-algebraic way to approach this question. First, given a fixed hypersurface, we show how one can determine whether points on the hypersurface determine it by looking at a corresponding marking of the dual complex. With a regularity condition on the dual complex and when the number of points is minimal, we show that our condition is equivalent to the connectedness of an appropriate sub-complex. Second, we introduce notions of non-singularity of tropical matrices and solutions to tropical linear equations that take into account our notion of multiplicity and prove a Cramer's Rule type theorem relating them.