非线性q-差分方程的Maillet-Malgrange定理的超度量形式
An ultrametric version of the Maillet-Malgrange theorem for nonlinear q-difference equations
摘要：
证明了关于非线性解析q-差分方程形式解的Gevrey性质的Maillet-Malgrange定理的一个超度量q-差分形式.由于°_q和\ord_q定义了{ \mathbb C}（q）上的两个赋值，特别地，当q为参数时，我们得到了非线性q-差分方程形式解在q上的次数增长和在q上的阶的一个结果.我们通过考虑两个例子来说明主要定理：对于非线性情况，"Painleve“II”的q-变形，以及在线性情况下，图8节点的有色琼斯多项式满足的q-差分方程。我们还考虑一个q模拟的Maillet-Malgrange定理，无论是在复杂的和在超度量设置，假设|q| =1和一个经典的丢番图条件。
英文摘要：
We prove an ultrametric q-difference version of the Maillet-Malgrange theorem, on the Gevrey nature of formal solutions of nonlinear analytic q-difference equations. Since °_q and \ord_q define two valuations on {\mathbb C}(q), we obtain, in particular, a result on the growth of the degree in q and the order at q of formal solutions of nonlinear q-difference equations, when q is a parameter. We illustrate the main theorem by considering two examples: a q-deformation of ``Painleve' II'', for the nonlinear situation, and a q-difference equation satisfied by the colored Jones polynomials of the figure 8 knots, in the linear case. We consider also a q-analog of the Maillet-Malgrange theorem, both in the complex and in the ultrametric setting, under the assumption that |q|=1 and a classical diophantine condition.
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