The Indefinite Summation Problem for the Laurent Ring
Abstract
This article solves the Indefinite Summation Problem (ISP) for the difference ring $(A, α)$, where A is the Laurent ring of shift operators on the lattice Zn, and α is any ring automorphism of A of finite order. The space of summable elements in $(A, α)$ is infinite dimensional, and it is possible to write down an explicit basis for it. The solution translates to a finite procedure involving a matrix multiplication, where the size of the matrix can be estimated. It follows that the arithmetic complexity of the solution can also be determined. These results extend to a solution of the ISP for the ring of functions on Zn, on which α acts by duality. The article also points out that the solution to the ISP amounts to calculating the group cohomologies Hi([α], A), i = 0, 1, where [α] is the cyclic group generated by α.
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BibTeX
@misc{shankar2026indefinite,
title = {{The Indefinite Summation Problem for the Laurent Ring}},
author = {Shiva Shankar},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2609.00824},
eprint = {2609.00824},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.00824},
}
RIS
TY - UNPB
TI - The Indefinite Summation Problem for the Laurent Ring
AU - Shiva Shankar
PY - 2026
DA - 2026/10/20
PB - arXiv
DO - 10.48550/arxiv.2609.00824
UR - https://arxiv.org/abs/2609.00824
AB - This article solves the Indefinite Summation Problem (ISP) for the difference ring $(A, α)$, where A is the Laurent ring of shift operators on the lattice Z^n, and α is any ring automorphism of A of finite order. The space of summable elements in $(A, α)$ is infinite dimensional, and it is possible to write down an explicit basis for it. The solution translates to a finite procedure involving a matrix multiplication, where the size of the matrix can be estimated. It follows that the arithmetic complexity of the solution can also be determined. These results extend to a solution of the ISP for the ring of functions on Z^n, on which α acts by duality. The article also points out that the solution to the ISP amounts to calculating the group cohomologies H^i([α], A), i = 0, 1, where [α] is the cyclic group generated by α.
ER -