Denominator Bounds and Polynomial Solutions for Systems of q-Recurrences over K(t) for Constant K

We consider systems A_\ell(t) y(q^\ell t) + … + A_0(t) y(t) = b(t) of higher order q-recurrence equations with rational coefficients. We extend a method for finding a bound on the maximal power of t in the denominator of arbitrary rational solutions y(t) as well as a method for bounding the degree of polynomial solutions from the scalar case to the systems case. The approach is direct and does not rely on uncoupling or reduction to a first order system. Unlike in the scalar case this usually requires an initial transformation of the system.

💡 Research Summary

The paper addresses the problem of determining explicit bounds on the denominator and the degree of polynomial solutions for systems of higher‑order q‑recurrence equations over the rational function field K(t), where K is a constant field of characteristic zero and q∈K{0} is not a root of unity. A system is written in the compact operator form

 A_s σ^s(y) + A_{s‑1} σ^{s‑1}(y) + ⋯ + A_0 y = b,

with A_i(t)∈K