Discrete Integrable Systems Associated with Orthogonal Polynomials on Algebraic Curves

Abstract

We extend the integrability framework originally introduced by Spicer, Nijhoff, and van der Kamp by reformulating it in terms of two real variables defined on the unit circle. This geometric perspective is then generalized to other algebraic curves, specifically, the Folium of Descartes and Viviani’s curve. Within this extended setting, we apply determinant identities to orthogonal polynomials and employ Christoffel and Geronimus transformations to construct discrete integrable systems associated with these curves. For each curve, we derive corresponding Lax pairs and matrix representations, highlighting the underlying symmetry and algebraic structure. The unit-circle case is closely related to the higher analogue of the discrete-time Toda equation (HADT) and the quotient-quotient-difference (QQD) scheme, while the systems associated with the Folium of Descartes and Viviani’s curve appear to provide new examples within the present framework. Our results illustrate how algebraic geometry and orthogonal polynomial theory can be effectively combined to construct discrete integrable dynamics, offering promising directions for further exploration in discrete integrability and geometric methods in mathematical physics.