{
  "abstract": "In this paper, we build upon the analysis initiated by Gennadiy Averkov and Matthias Schymura (2022) and establish that the number of distinct columns of a $Δ$-modular matrix $A \\in \\mathbb{Z}^{m \\times n}$ of rank $m$ is $O(m^3 Δ)$. This upper bound was previously known only for odd values of $Δ$. Recall that a matrix is called $Δ$-modular if the maximum of the absolute values of its $m \\times m$ minors equals $Δ$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2608.06215",
  "author_total": 3,
  "authors": [
    {
      "name": "Elizaveta Pribytkova"
    },
    {
      "name": "Dmitry Gribanov"
    },
    {
      "name": "Stanislav Moiseev"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/4fpn2rq8dhrwa.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2608.06215",
  "fulltext_url": null,
  "id": "4fpn2rq8dhrwa",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2608.06215",
  "language": null,
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  "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.CG"
    ],
    [
      "arxiv",
      "cs.DM"
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    [
      "arxiv",
      "math.CO"
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      "arxiv",
      "math.OC"
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      "openalex",
      "1703|Complexity and Algorithms in Graphs"
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  "openalex": "W7201866564",
  "pdf_url": "https://arxiv.org/pdf/2608.06215",
  "pmcid": null,
  "pmid": null,
  "published": "2026-08-29",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/4fpn2rq8dhrwa.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:17:48Z",
      "native_id": "2608.06215",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2608.06215"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.OC",
      "subfield": "optimisation-control"
    }
  ],
  "title": "Column Number of Delta-modular matrices: Refined Analysis via Sauer Matrices",
  "title_text": "Column Number of Delta-modular matrices: Refined Analysis via Sauer Matrices",
  "type": "preprint",
  "updated_at": "2026-11-04T15:20:51Z",
  "url": "https://research.draconitic.com/paper/4fpn2rq8dhrwa",
  "venue": "arXiv",
  "versions": []
}