Burchnall-Chaundy polynomials and the Laurent phenomenon

TL;DR

This paper explores the polynomial solutions of Burchnall-Chaundy polynomials and related difference equations, revealing their Laurent property and connections to integrable systems like KdV reductions and Hirota-Miwa equations.

Contribution

It extends the understanding of Burchnall-Chaundy polynomials by linking them to Laurent phenomena and integrable difference equations, providing new forms and insights.

Findings

Polynomial solutions are shown to be Laurent in initial data

Connections established between Burchnall-Chaundy polynomials and KdV-type reductions

New forms of polynomials expressed in terms of initial data

Abstract

The Burchnall-Chaundy polynomials $P_n(z)$ are determined by the differential recurrence relation $$P_{n+1}'(z)P_{n-1}(z)-P_{n+1}(z)P_{n-1}'(z)=P_n(z)^2$$ with $P_{-1}=P_0(z)=1.$ The fact that this recurrence relation has all solutions polynomial is not obvious and is similar to the integrality of Somos sequences and the Laurent phenomenon. We discuss this parallel in more detail and extend it to two difference equations $$Q_{n+1}(z+1)Q_{n-1}(z)-Q_{n+1}(z)Q_{n-1}(z+1)=Q_n(z)Q_n(z+1)$$ and $$R_{n+1}(z+1)R_{n-1}(z-1)-R_{n+1}(z-1)R_{n-1}(z+1)=R^2_n(z)$$ related to two different KdV-type reductions of the Hirota-Miwa and Dodgson octahedral equations. As a corollary we have a new form of the Burchnall-Chaundy polynomials in terms of the initial data $P_n(0)$, which is shown to be Laurent.