Supertropical semirings and supervaluations

TL;DR

This paper develops a theory of supertropical semirings and supervaluations, providing a lattice structure for supervaluations covering a valuation, and applies it to tropical geometry with a supertropical version of Kapranov's lemma.

Contribution

It introduces supertropical semirings and supervaluations, describes the lattice of supervaluations covering a valuation, and applies the theory to tropical geometry, including a supertropical Kapranov's lemma.

Findings

The set of supervaluations covering a valuation forms a complete lattice.

Explicit description of supervaluations covering a Krull valuation on a field.

A supertropical version of Kapranov's lemma enhances tropical geometry tools.

Abstract

We interpret a valuation $v$ on a ring $R$ as a map $v: R \to M$ into a so called bipotent semiring $M$ (the usual max-plus setting), and then define a \textbf{supervaluation} $\phi$ as a suitable map into a supertropical semiring $U$ with ghost ideal $M$ (cf. [IR1], [IR2]) covering $v$ via the ghost map $U \to M$. The set $\Cov(v)$ of all supervaluations covering $v$ has a natural ordering which makes it a complete lattice. In the case that $R$ is a field, hence for $v$ a Krull valuation, we give a complete explicit description of $\Cov(v)$. The theory of supertropical semirings and supervaluations aims for an algebra fitting the needs of tropical geometry better than the usual max-plus setting. We illustrate this by giving a supertropical version of Kapranov's lemma.