$q$ -完整序列的次数是二次拟多项式
The degree of a $q$-holonomic sequence is a quadratic quasi-polynomial
摘要：
变量$q$中的有理函数序列是$q$ -完整的，如果它满足系数多项式在$q$和$q^n$中的线性递归。证明了$q$ -完整序列的次数最终是二次拟多项式。我们的证明使用微分伽罗瓦论（适应证明有关完整的$D$ -模块的情况下，$q$ -完整的$D$ -模块）结合Lech-Mahler-Skolem定理从数论。利用线性$q$ -差分方程的牛顿多边形，在$q=0$上引入正则奇异$q$ -差分方程的概念和线性$q$ -差分方程解的WKB基.然后我们利用Lech-Mahler-Skolem定理来研究它们的首项的消失。与$q=1$的情形不同，WKB解的收敛性不存在解析问题，我们的证明是构造性的，并通过一个明确的例子加以说明。
英文摘要：
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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