Kernels in tropical geometry and a Jordan-H\"older Theorem

TL;DR

This paper develops a new algebraic framework using kernels and chains of homomorphisms in tropical geometry, leading to a tropical Jordan-H"older theorem and insights into tropical roots.

Contribution

It introduces a kernel-based algebraic approach in tropical geometry, extending classical chain and composition theories to the tropical setting.

Findings

Established a correspondence between $1^ u$-sets and $ u$-kernels of the rational semifield.

Developed a tropical Jordan-H"older theorem based on composition series and convexity degree.

Applied the theory to study tropical roots and their algebraic properties.

Abstract

A correspondence exists between affine tropical varieties and algebraic objects, following the classical Zariski correspondence between irreducible affine varieties and the prime spectrum of the coordinate algebra in affine algebraic geometry. Although in this context the natural analog of the polynomial ring over a field is the polynomial semiring over a semifield (without a zero element), one obtains homomorphic images of coordinate algebras via congruences rather than ideals, which complicates the algebraic theory considerably. In this paper, we pass to the semifield $F(\lambda_1, \dots, \lambda_n)$ of fractions of the polynomial semiring, for which there already exists a well developed theory of kernels, which are normal convex subgroups; this approach enables us to switch the structural roles of addition and multiplication and makes available much of the extensive theory of chains of homomorphisms of groups, including the Jordan-Holder theory. The parallel of the zero set now is the 1-set. These notions are refined in the language of supertropical algebra to $\nu$-kernels and $1^\nu$-sets, lending more precision to the theory. In analogy to Hilbert's celebrated Nullstellensatz which provides a correspondence between radical ideals and zero sets, we develop a correspondence between $1^\nu$-sets and a well-studied class of $\nu$-kernels of the rational semifield called polars, originating from the theory of lattice-ordered groups. This correspondence becomes simpler and more applicable when restricted to a special kind of kernel, called principal, intersected with the kernel generated by $F$. We utilize this theory to study tropical roots in tropical geometry. As an application, we develop composition series and convexity degree, leading to a tropical version of the Jordan-H\"{o}lder theorem.