{
  "abstract": "We study how the ranks of factors in a positive semidefinite slack factorization constrain its size. We bound the number of low-rank vertex factors in any fixed factorization, and use this to determine the asymptotic order of the minimum factorization size under any fixed bound on vertex-factor ranks: $Θ(n\\log n)$ for permutahedra $Π_n$ and $Θ(\\log N)$ for regular $N$-gons. In the rank-one case this gives the order of the minimum dimension of a space of real functions on the vertices in which every facet slack is a sum of squares, with no symmetry or degree restriction. For unrestricted lifts, the bound gives $\\mathrm{xc}_{\\mathrm{PSD}}(Π_n)\\ge n+\\log_3 n-O(1)$, and shows that a permutahedron factorization of linear size would need vertex factors of rank $Ω(\\log n)$ at all but a vanishing fraction of vertices. We also prove that every polytope of real positive semidefinite rank at most four has at most twelve vertices, and construct explicit size-four lifts for every square-symmetric octagon, so eight is attained. The upper bound combines a count of the polygon corners reached by curves of rank-one factors with incidence constraints on factor ranks. Whether eight is the maximum remains open.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.04498",
  "author_total": 1,
  "authors": [
    {
      "name": "Avinash Bhardwaj"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/4qjv9a01ymh60.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.04498",
  "fulltext_url": null,
  "id": "4qjv9a01ymh60",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.04498",
  "language": null,
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.CO"
    ],
    [
      "arxiv",
      "math.OC"
    ],
    [
      "openalex",
      "2612|Advanced Optimization Algorithms Research"
    ]
  ],
  "openalex": "W7220745421",
  "pdf_url": "https://arxiv.org/pdf/2610.04498",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-21",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/4qjv9a01ymh60.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:34:22Z",
      "native_id": "2610.04498",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.04498"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.OC",
      "subfield": "optimisation-control"
    }
  ],
  "title": "Rank-sensitive vertex bounds for semidefinite lifts",
  "title_text": "Rank-sensitive vertex bounds for semidefinite lifts",
  "type": "preprint",
  "updated_at": "2026-10-30T12:34:22Z",
  "url": "https://research.draconitic.com/paper/4qjv9a01ymh60",
  "venue": "arXiv",
  "versions": []
}