{
  "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 $\\Z^n$, 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 $α$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.00824",
  "author_total": 1,
  "authors": [
    {
      "name": "Shiva Shankar"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/bagcezw8bsydn.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.00824",
  "fulltext_url": null,
  "id": "bagcezw8bsydn",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.00824",
  "language": null,
  "licence": "LicenseRef-arXiv-nonexclusive-1.0",
  "licence_label": "arXiv licence",
  "licence_source": "article",
  "licence_url": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "cs.SC"
    ],
    [
      "arxiv",
      "math.OC"
    ],
    [
      "openalex",
      "2602|Rings, Modules, and Algebras"
    ]
  ],
  "openalex": "W7206210583",
  "pdf_url": "https://arxiv.org/pdf/2609.00824",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-20",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/bagcezw8bsydn.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:34:08Z",
      "native_id": "2609.00824",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.00824"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.OC",
      "subfield": "optimisation-control"
    }
  ],
  "title": "The Indefinite Summation Problem for the Laurent Ring",
  "title_text": "The Indefinite Summation Problem for the Laurent Ring",
  "type": "preprint",
  "updated_at": "2026-10-30T12:34:08Z",
  "url": "https://research.draconitic.com/paper/bagcezw8bsydn",
  "venue": "arXiv",
  "versions": []
}