{
  "abstract": "Let $I_h(a,b)$ be the ideal generated by the $h\\times h$ permanents of a generic $a\\times b$ matrix over a field of characteristic different from two. We prove that $\\dim K[X]/I_h(a,b)=\\max(a,b)(h-1)$ whenever $1\\le h\\le\\min(a,b)$ and $h<\\max(a,b)$. Consequently, the critical locus of the permanent of order $n$ has codimension $2n$ for every $n\\ge 2$. We also prove that the maximal permanents of a generic $k\\times(k+1)$ matrix generate a geometrically reduced complete intersection. The dimension argument uses a pivot expansion and cubic relations in the border variables. Two successive comparisons with highest-degree equations reduce the relevant fibers to a coordinate-support calculation. Reducedness follows from a separate regular-sequence argument and a dimension estimate excluding components over the exceptional locus.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.27320",
  "author_total": 1,
  "authors": [
    {
      "name": "Ying Xie"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/bxgvy6ggx2xc6.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.27320",
  "fulltext_url": null,
  "id": "bxgvy6ggx2xc6",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.27320",
  "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.AC"
    ],
    [
      "arxiv",
      "math.AG"
    ],
    [
      "openalex",
      "2602|Commutative Algebra and Its Applications"
    ]
  ],
  "openalex": "W7214240627",
  "pdf_url": "https://arxiv.org/pdf/2609.27320",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-23",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/bxgvy6ggx2xc6.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:24:00Z",
      "native_id": "2609.27320",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.27320"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AC",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "Dimensions of permanental varieties in arbitrary size",
  "title_text": "Dimensions of permanental varieties in arbitrary size",
  "type": "preprint",
  "updated_at": "2026-10-30T12:24:00Z",
  "url": "https://research.draconitic.com/paper/bxgvy6ggx2xc6",
  "venue": "arXiv",
  "versions": [
    {
      "doi": "10.5281/zenodo.22901136",
      "pid": null,
      "relation": "published-as",
      "title": null,
      "venue": null
    }
  ]
}