The degree of a $q$-holonomic sequence is a quadratic quasi-polynomial

A sequence of rational functions in a variable $q$ is $q$-holonomic if it satisfies a linear recursion with coefficients polynomials in $q$ and $q^n$. We prove that the degree of a $q$-holonomic sequence is eventually a quadratic quasi-polynomial. Our proof uses differential Galois theory (adapting proofs regarding holonomic $D$-modules to the case of $q$-holonomic $D$-modules) combined with the Lech-Mahler-Skolem theorem from number theory. En route, we use the Newton polygon of a linear $q$-difference equation, and introduce the notion of regular-singular $q$-difference equation and a WKB basis of solutions of a linear $q$-difference equation at $q=0$. We then use the Lech-Mahler-Skolem theorem to study the vanishing of their leading term. Unlike the case of $q=1$, there are no analytic problems regarding convergence of the WKB solutions.Our proofs are constructive, and they are illustrated by an explicit example.

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