Hankel Determinant Solutions to Several Discrete Integrable Systems and the Laurent Property

Source 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 Bäcklund transformation of the Somos-4 recurrence. Related topics about Somos polynomials are also studied.