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In 2012 Chen and Singer introduced the notion of discrete residues for rational functions as a complete obstruction to rational summability. More explicitly, for a given rational function f(x), there exists a rational function g(x) such that f(x) = g(x+1) - g(x) if and only if every discrete residue of f(x) is zero. Discrete residues have many important further applications beyond summability: to creative telescoping problems, thence to the determination of (differential-)algebraic relations among hypergeometric sequences, and subsequently to the computation of (differential) Galois groups of difference equations. However, the discrete residues of a rational function are defined in terms of its complete partial fraction decomposition, which makes their direct computation impractical due to the high complexity of completely factoring arbitrary denominator polynomials into linear factors. We develop a factorization-free algorithm to compute discrete residues of rational functions, relying only on gcd computations and linear algebra.
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Twisted Mahler Discrete Residues
Abstract Recently we constructed Mahler discrete residues for rational functions and showed they comprise a complete obstruction to the Mahler summability problem of deciding whether a given rational function $f(x)$ is of the form $$g(x^{p})-g(x)$$ for some rational function $g(x)$ and an integer $p> 1$. Here we develop a notion of $$\lambda $$-twisted Mahler discrete residues for $$\lambda \in \mathbb{Z}$$, and show that they similarly comprise a complete obstruction to the twisted Mahler summability problem of deciding whether a given rational function $f(x)$ is of the form $$p^{\lambda } g(x^{p})-g(x)$$ for some rational function $g(x)$ and an integer $p>1$. We provide some initial applications of twisted Mahler discrete residues to differential creative telescoping problems for Mahler functions and to the differential Galois theory of linear Mahler equations.
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- Award ID(s):
- 1815108
- PAR ID:
- 10550413
- Publisher / Repository:
- Oxford University Press
- Date Published:
- Journal Name:
- International Mathematics Research Notices
- Volume:
- 2024
- Issue:
- 23
- ISSN:
- 1073-7928
- Format(s):
- Medium: X Size: p. 14259-14288
- Size(s):
- p. 14259-14288
- Sponsoring Org:
- National Science Foundation
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