Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Kernels in tropical geometry and a Jordan-H\"older Theorem   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.

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The categories of this paper are: solv-int
21
The categories of this paper are: adap-org
26
The categories of this paper are: bayes-an
3
The categories of this paper are: physics
32
The categories of this paper are: atom-ph
13
The categories of this paper are: quant-ph
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The categories of this paper are: funct-an
0
The categories of this paper are: hep-ph
18
The categories of this paper are: econ
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The categories of this paper are: gr-qc
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The categories of this paper are: ao-sci
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The categories of this paper are: hep-th
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The categories of this paper are: q-bio
1
The categories of this paper are: cs
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The categories of this paper are: alg-geom
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The categories of this paper are: chao-dyn