From discrete integrable equations to Laurent recurrences

We show how to obtain relations for the divisors of terms generated by a homogenized version of a rational recurrence. When the rational recurrence confines singularities the relations take the form of a rational recurrence, possibly with periodic coefficients. As the recurrence generates polynomials one expects it to possess the Laurent property. The method we develop uses ultra-discretization and recursive factorization. It is applied to certain QRT-maps which gives rise to Somos-k (k = 4, 5) sequences with periodic coefficients. Novel (N + 3)-rd order recurrences are obtained from the Nth order DTKQ-equation (N = 2, 3). In each case the resulting recurrence equation has the Laurent property. The method is equally applicable to non-integrable or non-confining equations. However, in the latter case the degree and the order of the relation might display unbounded growth. We demonstrate the difference, by considering different parameter choices in a generalized Lyness equation.