We analyze the differential equations produced by the method of creative telescoping applied to a hyperexponential term in two variables. We show that equations of low order have high degree, and that higher order equations have lower degree. More precisely, we derive degree bounding formulas which allow to estimate the degree of the output equations from creative telescoping as a function of the order. As an application, we show how the knowledge of these formulas can be used to improve, at least in principle, the performance of creative telescoping implementations, and we deduce bounds on the asymptotic complexity of creative telescoping for hyperexponential terms.

我们分析了在两个变量中应用到超指数项的创造性伸缩方法所产生的微分方程。我们表明，低阶方程具有较高的度数，而高阶方程具有较低的度数。更准确地说，我们推导了度数边界公式，该公式可以根据创意伸缩性来估计输出方程的阶次。作为一个应用程序，我们展示了如何至少在原则上使用这些公式的知识来改善创意伸缩实现的性能，并且我们为超指数项推论了创意伸缩的渐进复杂性的界限。