Diamond Determinants and Somos Sequences

TL;DR

This paper proves finite-rank properties of Somos sequences of orders 6 and 7 using elementary methods, extending known results and proposing conjectures for higher orders within Gale-Robinson sequences.

Contribution

It provides an elementary proof for the finite-rank property of Somos sequences of order 6 and introduces a new finite-rank result for order 7, expanding understanding of these sequences.

Findings

Finite-rank property established for Somos sequences of order 6.

New finite-rank property demonstrated for Somos sequences of order 7.

Conjectures proposed for higher-order Gale-Robinson sequences.

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.