Hankel determinant solutions to several discrete integrable systems and the Laurent Property

TL;DR

This paper derives Hankel determinant solutions for several discrete integrable systems, confirming their Laurent property and revealing relationships between different recurrences, advancing understanding of their algebraic structure.

Contribution

It introduces Hankel determinant solutions for the Somos-4, Somos-5, and related $A_1$ $Q$-system, demonstrating their Laurent property and connections among these systems.

Findings

Hankel determinant solutions confirm Laurent property for the systems.

Somos-5 recurrence is a Bäcklund transformation of Somos-4.

Analysis of Somos polynomials related to these systems.

Abstract

Many discrete integrable systems exhibit the Laurent phenomenon. In this paper, we investigate three integrable systems: the Somos-4 recurrence, the Somos-5 recurrence and a system related to so-called $A_1$ $Q$-system, whose general solutions are derived in terms of Hankel determinant. As a result, we directly confirm that they satisfy the Laurent property. Additionally, it is shown that the Somos-5 recurrence can be viewed as a specified B\"acklund transformation of the Somos-4 recurrence. The related topics about Somos polynomials are also studied.