Efficient computation of q-discrete residues without partial fractions

Develop efficient algorithms that, given a rational function f in the difference field (M,σ) with M=K(x) and σ(x)=qx for q∈K× not a root of unity, compute K-rational representations of the q-discrete residues of f at all orbits and orders while bypassing the computation of the complete partial fraction decomposition of f, analogous to the available methods in the shift case (σ(x)=x+1).

Background

In the q-dilation setting, summability of rational functions f∈K(x) with σ(x)=qx (q not a root of unity) is characterized by the vanishing of q-discrete residues, as introduced by Chen and Singer. While the theoretical characterization is known, practical computation of these residues is impeded by the need for the complete partial fraction decomposition, which can be expensive or infeasible.

For the shift case (σ(x)=x+1), recent work has produced efficient algorithms that generate K-rational representations of discrete residues without computing the full partial fraction decomposition. The paper explicitly notes the absence of analogous algorithms for the q-dilation case, highlighting a concrete open problem to bridge this computational gap.