Kernels in tropical geometry and a Jordan-Hölder theorem

Tal Perri, Louis H. Rowen

Research output: Contribution to journalArticlepeer-review

3 Scopus citations

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, 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,⋯, λn) 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, ⋯, λn); 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 1v-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 1v-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.

Original languageEnglish
Article number1850066
JournalJournal of Algebra and its Applications
Volume17
Issue number4
DOIs
StatePublished - 1 Apr 2018

Bibliographical note

Publisher Copyright:
© 2018 World Scientific Publishing Company.

Funding

The second author would like to thank the University of Virginia for its support during the preparation of this revision.

FundersFunder number
University of Virginia

    Keywords

    • Congruence
    • Jordan-Holder
    • Laurent polynomial
    • archimedean
    • completion
    • corner internal
    • corner locus
    • dimension
    • order
    • polar
    • polynomial
    • principal kernel
    • region
    • regular kernel
    • root
    • semifield
    • semifield kernel
    • semigroup
    • semiring
    • supertropical algebra
    • tropical algebra
    • tropical geometry
    • tropical hypersurface
    • wedge decomposition

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