Kernels in tropical geometry and a Jordan–Hölder theorem

Abstract

When considering affine tropical geometry, one often works over the max-plus algebra (or its supertropical analog), which, lacking negation, is a semifield (respectively, [Formula: see text]-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[Formula: see text][Formula: see text], we pass to the semifield[Formula: see text][Formula: see text] of fractions of the polynomial semiring[Formula: see text], for which there already exists a well developed theory of kernels, which are normal convex subgroups of [Formula: see text]; the parallel of the zero set now is the [Formula: see text]-set, the set of vectors on which a given rational function takes the value 1. These notions are refined in supertropical algebra to [Formula: see text]-kernels (Definition 4.1.4) and [Formula: see text]-sets, which take the place of tropical varieties viewed as sets of common ghost roots of polynomials. The [Formula: see text]-kernels corresponding to tropical hypersurfaces are the [Formula: see text]-sets of what we call “corner internal rational functions,” and we describe [Formula: see text]-kernels corresponding to “usual” tropical geometry as [Formula: see text]-kernels which are “corner-internal” and “regular.” This yields an explicit description of tropical affine varieties in terms of various classes of [Formula: see text]-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 [Formula: see text]-sets and a class of [Formula: see text]-kernels of the rational [Formula: see text]-semifield[Formula: see text] called polars, originating from the theory of lattice-ordered groups. When [Formula: see text] is the supertropical max-plus algebra of the reals, this correspondence becomes simpler and more applicable when restricted to principal [Formula: see text]-kernels, intersected with the [Formula: see text]-kernel generated by [Formula: see text]. 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 [Formula: see text]-kernels.