{
  "abstract": "We present an analogue of Brion's theorem for the projected integer-point transform. Brion's theorem writes the integer-point transform of a rational polyhedron $P$ as the sum of the rational functions associated to its vertex cones. The pushforward $π_*$ along a lattice projection $π$ turns the transform of $P$ into the generating function of the numbers of lattice points in the fibres of $π$. We group the vertices of $P$ into contracted classes, generated by the edges whose direction $π$ kills, and show that the sum of the rational functions associated to the vertex cones in a contracted class can be pushed forward along $π$, yielding a rational function decomposition of the projected integer-point transform. For a lattice polytope, the classes correspond to the connected components of the fixed locus of the associated toric variety under the action of a subtorus, and our formula resembles the localization formula for this subtorus. We use this decomposition to provide an explicit formula for the projected integer-point transform of a linear family of polytopes. We illustrate this result with families of Gelfand--Zetlin and string polytopes.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.38674",
  "author_total": 2,
  "authors": [
    {
      "name": "Matthew Dupraz"
    },
    {
      "name": "Leonid Monin"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/7f3wnwbjaxfeh.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.38674",
  "fulltext_url": null,
  "id": "7f3wnwbjaxfeh",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.38674",
  "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",
      "math.AG"
    ],
    [
      "arxiv",
      "math.CO"
    ],
    [
      "arxiv",
      "math.RT"
    ]
  ],
  "openalex": null,
  "pdf_url": "https://arxiv.org/pdf/2610.38674",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-28",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/7f3wnwbjaxfeh.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:36:32Z",
      "native_id": "2610.38674",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.38674"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AG",
      "subfield": "algebra-number-theory"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Projected Brion's theorem",
  "title_text": "Projected Brion's theorem",
  "type": "preprint",
  "updated_at": "2026-11-01T12:20:50Z",
  "url": "https://research.draconitic.com/paper/7f3wnwbjaxfeh",
  "venue": "arXiv",
  "versions": []
}