Generic Tropical Varieties on Subvarieties and in the Non-constant Coefficient Case

TL;DR

This paper extends the concept of generic tropical varieties to non-constant coefficients and arbitrary subvarieties, providing structural insights into how tropical varieties behave under linear coordinate changes.

Contribution

It generalizes the existence of generic tropical varieties beyond constant coefficients and arbitrary subvarieties, including algebraic groups, and explores their structural properties.

Findings

Existence of generic tropical varieties in non-constant coefficient cases.

Structural results on tropical varieties under linear coordinate changes.

Extension of genericity notions to algebraic subvarieties of the general linear group.

Abstract

In earlier papers it was shown that the generic tropical variety of an ideal can contain information on algebraic invariants as for example the depth in a direct way. The existence of generic tropical varieties has so far been proved in the constant coefficient case for the usual notion of genericity. In this paper we generalize this existence result to include the case of non-constant coefficients in certain settings. Moreover, we extend the notion of genericity to arbitrary closed subvarieties of the general linear group. In addition to including the concept of genericity on algebraic groups this yields structural results on the tropical variety of an ideal under an arbitrary linear coordinate change.