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arXiv · 2602.24239

Diamond Determinants and Somos Sequences

Nikolai Beluhov
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Abstract

A Somos sequence of order $n$ is defined by a quadratic recurrence of width $n + 1$. Some of the remarkable properties of these sequences for small $n$ are tied to certain matrices built out of them being of finite rank. We give an elementary proof of the finite-rank property for order $6$, previously only established with the help of advanced machinery from the theory of hyperelliptic functions. Our method also yields a new finite-rank property for the Somos sequences of order $7$. In addition, we conjecture generalisations of these results to higher orders, for the subclass of Gale-Robinson sequences.

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Nikolai Beluhov. 2026-02-27. Diamond Determinants and Somos Sequences. https://arxiv.org/abs/2602.24239

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