Tropicalization and irreducibility of Generalized Vandermonde Determinants

TL;DR

This paper investigates the conditions under which generalized Vandermonde determinants are irreducible over algebraic closures and when their tropicalizations remain irreducible, linking algebraic and tropical geometry.

Contribution

It provides new geometric and arithmetic criteria for irreducibility of generalized Vandermonde determinants and their tropicalizations.

Findings

Characterization of irreducibility conditions over algebraic closures

Criteria for tropicalization irreducibility of determinants

Linking algebraic properties with tropical geometry insights

Abstract

We find geometric and arithmetic conditions in order to characterize the irreducibility of the determinant of the generic Vandermonde matrix over the algebraic closure of any field k. We also characterize those determinants whose tropicalization with respect to the variables of a row is irreducible.