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Algebraic dependence of cluster X-variables

A mix of human-written and AI-generated textHuman understanding: some partsmath.RA — Rings and Algebrasmath.CO — Combinatorics

Contributed by Qiyue Tang

Version 2 / Oct 05, 2026 / CC BY 4.0

Abstract

We prove that, over any field and with algebraically independent initial seed coordinates, two cluster XX-variables are algebraically dependent if and only if they are equal or reciprocal. In characteristic zero, this is equivalent to the vanishing of the wedge product of their differentials.

Provenance statement

This problem originates from MathOverflow. Recently, I came across a post from several years ago that described the problem summarized in the abstract. I found the problem statement concise and quite intriguing. In a single conversation with GPT-6-Astra—lasting no more than thirty minutes—a satisfactory solution was reached. I then used the built-in Writing Skill to prepare the manuscript, and independently used GPT-6-Astra to verify the correctness of the paper twice.

References

  1. P. Cao, B. Keller and F. Qin, The valuation pairing on an upper cluster algebra , J. Reine Angew. Math. 806 (2024), 71--114. doi:10.1515/crelle-2023-0080 (https://doi.org/10.1515/crelle-2023-0080). arXiv:2204.09576v2 (https://arxiv.org/abs/2204.09576v2).arXivDOI
  2. Elbabak, Differential (in)dependancies between cluster variables , MathOverflow, question 302052, 5 June 2018. https://mathoverflow.net/q/302052 (accessed 5 October 2026).link
  3. V. V. Fock and A. B. Goncharov, Cluster ensembles, quantization and the dilogarithm , Ann. Sci. Éc. Norm. Supér. (4) 42 (2009), 865--930. doi:10.24033/asens.2112 (https://doi.org/10.24033/asens.2112).DOI
  4. M. Sherman-Bennett, Combinatorics of X\mathcal X-variables in finite type cluster algebras , J. Combin. Theory Ser. A 165 (2019), 273--298. arXiv:1803.02492v2 (https://arxiv.org/abs/1803.02492v2).arXiv

Version history

  1. v1 · 8 pagesSubmitted by 琪越 唐Initial depositSupersededOct 05, 2026
  2. v2 · 8 pagesSubmitted by Qiyue TangSubmitted as a new versionCurrentNov 06, 2026