On a certain system of discrete equations with the Laurent property

Abstract

We consider a system of two bilinear discrete equations with four parameters. It defines a certain discrete dynamical system. Each of its solutions is represented as a pair of sequences. It is proved that for arbitrary initial data all terms of both sequences are Laurent polynomials in the initial data with integer coefficients. This property is called the Laurent property. The proof relies on an established connection between the system and the Somos-5 equation. It is shown that both components of the solution satisfy the same equation with the same parameters. To prove the main theorem, the strong Laurent property for the Somos-5 equation is used, which makes it possible to control negative powers of the initial data. An analogue of this property is obtained directly for the system under study. As corollaries, sufficient conditions for the integrality of solutions are found. An example of a pair of integer sequences illustrating the theorem on the strong Laurent property is given.