# On the Existence of Telescopers for Rational Functions in Three Variables

## TL;DR

This paper addresses the fundamental problem of determining when telescopers exist for rational functions in three variables, providing criteria that enable effective algorithm termination in creative telescoping.

## Contribution

It reduces the existence problem from three variables to two, solving 18 cases and enabling efficient construction of telescopers for trivariate rational functions.

## Key findings

- Reduced the existence problem from three to two variables.
- Provided criteria for telescoper existence in 18 cases.
- Enabled termination detection for algorithms with trivariate inputs.

## Abstract

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.