Plea for diagonals and telescopers of rational functions

TL;DR

This paper advocates for the use of creative telescoping to analyze diagonals and telescopers of rational and algebraic functions, revealing invariance properties and homomorphism relations that deepen understanding of their algebraic structure.

Contribution

It demonstrates invariance of diagonals under birational transformations and explores the homomorphic nature of telescopers, extending to complex examples like hypergeometric functions and elliptic curves.

Findings

Diagonals of rational functions are invariant under certain birational transformations.

Telescopers of rational functions are often homomorphic to their adjoints.

Examples include hypergeometric functions related to elliptic curves and Ising model correlations.

Abstract

This paper is a plea for diagonals and telescopers of rational, or algebraic, functions using creative telescoping, in a computer algebra experimental mathematics learn-by-examples approach. We show that diagonals of rational functions (and this is also the case with diagonals of algebraic functions) are left invariant when one performs an infinite set of birational transformations on the rational functions. These invariance results generalize to telescopers. We cast light on the almost systematic property of homomorphism to their adjoint of the telescopers of rational, or algebraic, functions. We shed some light on the reason why the telescopers, annihilating the diagonals of rational functions of the form P/Q^k and 1/Q, are homomorphic. For telescopers with solutions (periods) corresponding to integration over non-vanishing cycles, we have a slight generalization of this result. We introduce some challenging examples of generalization of diagonals of rational functions, like diagonals of transcendental functions, yielding simple $_2F_1$ hypergeometric functions associated with elliptic curves, or (differentially algebraic) lambda-extension of correlation of the Ising model.