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Positive bases and polytopes for Laurent phenomenon algebras

Primarily AI-generated textHuman understanding: some partsmath.CO — Combinatoricsmath.AC — Commutative Algebramath.RA — Rings and Algebras

Contributed by Qiyue Tang ↗

Version 1 / Nov 11, 2026 / CC BY 4.0

Abstract

We prove that the cluster monomials of every graph Laurent phenomenon algebra form an atomic positive basis. For every finite normalized Laurent phenomenon algebra over a unique factorization domain, we construct a simple lattice polytope with unimodular normal fan whose polar realizes the cluster complex. We also prove that cluster monomials form a basis of the upper algebra in the finite case. In particular, every linear Laurent phenomenon algebra admits such a polyhedral realization.

Author comments: By posting this manuscript on this platform, I do not seek to assert ownership of the results or claim priority. I welcome anyone interested in preparing a finished account or checking the arguments; the eventual scholarly credit should go to those who undertake that work.

Provenance statement

GPT-6-Pro was used for a total of 7 hours of mathematical research and proof development. GPT-6-Pro was also used for independent proof checking, critical review, manuscript revision, and language polishing. An additional independent review by Rethlas-OSV is planned and will be reported separately.

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Version history

  1. v1 · 26 pagesSubmitted by Qiyue TangBy posting this manuscript on this platform, I do not seek to assert ownership of the results or claim priority. I welcome anyone interested in preparing a finished account or checking the arguments; the eventual scholarly credit should go to those who undertake that work.CurrentNov 10, 2026