Zeilberger's method of creative telescoping is crucial for the computer-generated proofs of combinatorial and special-function identities. Telescopers are linear differential or ($q$-)recurrence operators computed by algorithms for creative telescoping. For a given class of inputs, when telescopers exist and how to construct telescopers efficiently if they exist are two fundamental problems related to creative telescoping. In this paper, we solve the existence problem of telescopers for rational functions in three variables including 18 cases. We reduce the existence problem from the trivariate case to the bivariate case and some related problems. The existence criteria given in this paper enable us to determine the termination of algorithms for creative telescoping with trivariate rational inputs.

中文翻译：

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三变量有理函数望远镜的存在性

Zeilberger 的创造性伸缩方法对于组合和特殊功能身份的计算机生成证明至关重要。Telescopers 是线性微分或 ($q$-) 递归运算符，由用于创造性伸缩的算法计算。对于给定类别的输入，望远镜何时存在以及如果存在望远镜如何有效地构建望远镜是与创造性望远镜相关的两个基本问题。在本文中，我们解决了三变量有理函数望远镜的存在性问题，包括18种情况。我们将存在问题从三变量情况简化为双变量情况以及一些相关问题。本文中给出的存在标准使我们能够确定具有三变量理性输入的创造性伸缩算法的终止。