Year: 2026
Type: preprint
Abstract: We study tropical rational functions on the tropical projective line with three zeros and three poles in the torus, counted with multiplicity, modulo target automorphisms. The principal divisor identifies their moduli space with unordered pairs of disjoint effective divisors of degree three. We classify its \(53\) cells and prove that it is six-dimensional with ten contractible connected components. Source--target automorphism groups are trivial or of order two, and reflection loci are defined by linear gap equations. Allowing common support and points at infinity gives a divisor-pair compactification. Over an algebraically closed non-Archimedean field with value group \(\mathbb{R}\), root--pole tropicalization is surjective from valuation-separated cubic rational functions modulo the target torus normalizer. Explicit counterexamples show that the valuation of the conjugacy invariant \(J^2/I^3\) does not descend to this tropical moduli space. Generic functions admit minimal six-unit shallow ReLU representations with three positive and three negative unit output weights.
Source: arXiv (Cornell University)
Authors: Shaska, Tony, Siadat, Mohammad-Reza
Institution: Oakland University
Language: English
Subfield: Geometry and Topology
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7161555314