## 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(λ_{1},⋯, 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(λ_{1}, ⋯, 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.

Original language | English |
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Article number | 1850066 |

Journal | Journal of Algebra and its Applications |

Volume | 17 |

Issue number | 4 |

DOIs | |

State | Published - 1 Apr 2018 |

### Bibliographical note

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

Publisher Copyright:

© 2018 World Scientific Publishing Company.

## 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