Non-minimality of minimal telescopers explained by residues
Shaoshi Chen, Manuel Kauers, Christoph Koutschan, Xiuyun Li, Rong-Hua Wang, Yisen Wang
TL;DR
Addresses why the minimal telescoper is not always the minimal-order annihilator in definite summation and integration. Develops a residues-based, module-theoretic approach that links residue obstructions to $LCLM$-based structure and prescopers for hypergeometric terms, with a kernel-submodule/zero-sum framework to explain non-minimality. Key contributions include a detailed treatment of residues in rational functions, a practical prescoper construction, automorphism analysis of kernel submodules, and explicit vanishing-sum arguments. The framework enhances understanding of creative telescoping and provides computational tools to obtain truly minimal operators and diagnose overshoot phenomena.
Abstract
Elaborating on an approach recently proposed by Mark van Hoeij, we continue to investigate why creative telescoping occasionally fails to find the minimal-order annihilating operator of a given definite sum or integral. We offer an explanation based on the consideration of residues.