Twisted Mahler discrete residues
Carlos E. Arreche, Yi Zhang
TL;DR
This work develops λ-twisted Mahler discrete residues as a complete obstruction to λ-Mahler summability for rational functions over an algebraically closed field of characteristic zero. By decomposing f into ∞-, non-torsion, and torsion-tree components and introducing cycle maps 𝒟_{λ,τ} and ω-sections, the authors derive closed-form residue criteria dres_λ(f,∞), dres_λ(f,τ,k) whose vanishing exactly characterizes summability. The main theorem unifies the ∞-, non-torsion, and torsion analyses and leads to a global λ-Mahler reduction bar{f}_λ, which isolates the summable part from a canonical remainder. These results have practical impact on differential creative telescoping for Mahler functions and the differential Galois theory of linear Mahler equations, providing obstruction-based tools complementary to certificate-based algorithms.
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 $λ$-twisted Mahler discrete residues for $λ\in\mathbb{Z}$, 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 $p^λg(x^p)-g(x)$ 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.