Algebraic Aspects in Tropical Mathematics

Abstract

Much like in the theory of algebraic geometry, we develop a correspondence between certain types of algebraic and geometric objects. The basic algebraic environment we work in is the a semifield of fractions H(x1,...,xn) of the polynomial semidomain H[x1,...,xn], where H is taken to be an idempotent semifield, while for the geometric environment we have the space H^n. We show that taking H to be idempotent makes both H(x1,...,xn) and Hn idempotent which turn out to satisfy many desired properties that we utilize for our construction. The fundamental algebraic and geometric objects having interrelations are called kernels, encapsulating congruences over semifields and skeletons which serve as the analog for zero-sets of algebraic geometry. As an analog for the celebrated Nullstellenzats theorem, we develop a correspondence between skeletons and a family of kernels called polars originally developed in the theory of lattice-ordered groups. For a special kind of skeletons, called principal skeletons, we have simplified the correspondence by restricting our algebraic environment to a certain kernel of H(x1,...,xn). After establishing the linkage between kernels and skeletons we proceed to construct a second linkage, this time between a family of skeletons and what we call "corner-loci". Essentially a corner locus is what is called a tropical variety in the theory of tropical geometry, which is a set of corner roots of some set of tropical polynomials. A skeleton and a corner-locus define the exact same subset of H^n in different ways: while a corner locus is defined by corner roots of tropical polynomials, the skeleton is defined by equating fractions from H(x1,...,xn) to 1. All the connections presented above form a path connecting a tropical variety to a certain kind of kernel.