Equations of tropical varieties

TL;DR

This paper develops a scheme-theoretic framework for tropical geometry, defining tropical hypersurfaces as schemes over idempotent semirings and establishing a tropicalization functor that preserves key algebraic properties.

Contribution

It introduces a scheme-theoretic approach to tropical geometry using semiring schemes, extending tropicalization to a functorial setting that preserves multiplicities and Hilbert polynomials.

Findings

Tropical hypersurfaces are realized as solutions to explicit tropical equations.

The tropicalization functor is compatible with scheme structures and valuations.

Hilbert polynomial is preserved under tropicalization for projective subschemes.

Abstract

We introduce a scheme-theoretic enrichment of the principal objects of tropical geometry. Using a category of semiring schemes, we construct tropical hypersurfaces as schemes over idempotent semirings such as $\mathbb{T} = (\mathbb{R}\cup \{-\infty\}, \mathrm{max}, +)$ by realizing them as solution sets to explicit systems of tropical equations that are uniquely determined by idempotent module theory. We then define a tropicalization functor that sends closed subschemes of a toric variety over a ring R with non-archimedean valuation to closed subschemes of the corresponding tropical toric variety. Upon passing to the set of $\mathbb{T}$-points this reduces to Kajiwara-Payne's extended tropicalization, and in the case of a projective hypersurface we show that the scheme structure determines the multiplicities attached to the top-dimensional cells. By varying the valuation, these tropicalizations form algebraic families of $\mathbb{T}$-schemes parameterized by a moduli space of valuations on R that we construct. For projective subschemes, the Hilbert polynomial is preserved by tropicalization, regardless of the valuation. We conclude with some examples and a discussion of tropical bases in the scheme-theoretic setting.