Quadratic Equations in Tropical Regions

Abstract

In this note, the reader is invited to a walk through tropical semifields and the places where they border on
“ordinary” algebra. Though mostly neglected in today’s lectures on algebra, we point to the places where tropical
structures inevitably pervade, and show that they frequently occur in ring theory and classical algebra, touching at
least functional analysis, and algebraic geometry. Specifically, it is explained how valuation theory, which plays
an essential part in classical commutative algebra and algebraic geometry, is essentially tropical. In particular, it
is shown that Eisenstein’s well-known irreducibility criterion and other more powerful criteria follow immediately
by tropicalization. Some applications to algebraic equations in characteric 1, neat B´ezout domains, and rings of
continuous functions are given.