Kernels in tropical geometry and a Jordan-Hölder theorem

When considering affine tropical geometry, one often works over the max-plus algebra (or its supertropical analog), which, lacking negation, is a semifield (respectively, v-semifield) rather than a field. One needs to utilize congruences rather than ideals, leading to a considerably more complicated theory. In his dissertation, the first author exploited the multiplicative structure of an idempotent semifield, which is a lattice ordered group, in place of the additive structure, in order to apply the extensive theory of chains of homomorphisms of groups. Reworking his dissertation, starting with a semifield† F, we pass to the semifield†F(λ₁,⋯, λ_n) of fractions of the polynomial semiring†, for which there already exists a well developed theory of kernels, which are normal convex subgroups of F(λ₁, ⋯, λ_n); the parallel of the zero set now is the 1-set, the set of vectors on which a given rational function takes the value 1. These notions are refined in supertropical algebra to v-kernels (Definition 4.1.4) and 1^v-sets, which take the place of tropical varieties viewed as sets of common ghost roots of polynomials. The v-kernels corresponding to tropical hypersurfaces are the 1v-sets of what we call "corner internal rational functions," and we describe v-kernels corresponding to "usual" tropical geometry as v-kernels which are "corner-internal" and "regular." This yields an explicit description of tropical affine varieties in terms of various classes of v-kernels. The literature contains many tropical versions of Hilbert's celebrated Nullstellensatz, which lies at the foundation of algebraic geometry. The approach in this paper is via a correspondence between 1^v-sets and a class of v-kernels of the rational v-semifield† called polars, originating from the theory of lattice-ordered groups. When F is the supertropical max-plus algebra of the reals, this correspondence becomes simpler and more applicable when restricted to principal v-kernels, intersected with the v-kernel generated by F. For our main application, we develop algebraic notions such as composition series and convexity degree, leading to a dimension theory which is catenary, and a tropical version of the Jordan-Hölder theorem for the relevant class of v-kernels.