Bivariate Extensions of Abramov’s Algorithm for Rational Summation

Abstract

Abramov's algorithm enables us to decide whether a univariate rational function can be written as a difference of another rational function, which has been a fundamental algorithm for rational summation. In 2014, Chen and Singer generalized Abramov's algorithm to the case of rational functions in two (q-)discrete variables. In this paper we solve the remaining three mixed cases, which completes our recent project on bivariate extensions of Abramov's algorithm for rational summation.

Existence Problem of Telescopers for Rational Functions in Three Variables

“…m Tr L/F(y) (u i ) ∈ F(y). The exactness problem for bivariate rational functions can be determined by reductions (see [13,15,24,8]). For a later convenience, we summarize these results below.…”

Mahler Discrete Residues and Summability for Rational Functions

“…Continuous residues are fundamental and crucial tools in complex analysis, and have extensive and compelling applications in combinatorics [16]. In the last decade, a theory of (q-)discrete residues was proposed in [14] for the study of telescoping problems, which has found essential applications in several other closely related problems (see [3,4,11,19] for some examples). A theory of residues for skew rational functions was developed in [9], and then extended to Ore polynomials and applied to linearized Reed-Solomon codes in [10].…”

“…Continuous residues are fundamental and crucial tools in complex analysis, and have extensive and compelling applications in combinatorics [FS09]. In the last decade, a theory of discrete and q-discrete residues was proposed in [CS12] for the study of telescoping problems for bivariate rational functions, and subsequently found applications in the computation of differential Galois groups of second-order linear difference [Arr17] and q-difference equations [AZ22a] and other closely-related problems [Che18,HW15]. More recently, the authors of [Car21,CD23] developed a theory of residues for skew rational functions, which has important applications in duals of linearized Reed-Solomon codes [CD23].…”

An Algorithmic Approach to the $q$-Summability Problem of Bivariate Rational Functions

“…However, they did not provide a complete answer to the summability problem of bivariate functions. The first necessary and sufficient condition for the summability of bivariate functions was presented by Chen and Singer [8] for the rational case, and later extended to the remaining mixed cases by Chen in [5]. Based on the theoretical criterion given in [8], Hou and the author [11] presented a new criterion and an algorithm for deciding the summability of bivariate rational functions.…”