{
  "abstract": "We study the generic nullity of generalized commutator operators \\[L_{\\mathbf{A}}(X)=s_{k+1}(A_1,\\cdots,A_k,X)\\] on matrix algebras, where $s_{k+1}$ denotes the standard polynomial. Dixon and Pressman conjectured an explicit formula for the generic nullity of $L_{\\mathbf{A}}$, and Brassil and Reichstein proved the conjecture when $k$ is even. In this paper, we settle the remaining case where $k$ is odd. Our proof first treats the boundary cases $k=2n-3$ in dimensions $n$ and $n+1$ using degree decompositions and graph-theoretic interpretations of alternating trace forms, and then establishes a dimension-extension argument from $n$ to $n+2$. Consequently, together with the result of Brassil and Reichstein, we obtain a complete proof of the Dixon-Pressman generic nullity conjecture over any field of characteristic zero.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.06339",
  "author_total": 3,
  "authors": [
    {
      "name": "Chen Lin"
    },
    {
      "name": "Chenhao Tang"
    },
    {
      "name": "Enhan Zhao"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/8wgc5yxn54vgp.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.06339",
  "fulltext_url": null,
  "id": "8wgc5yxn54vgp",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.06339",
  "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.CO"
    ],
    [
      "arxiv",
      "math.RA"
    ],
    [
      "openalex",
      "2602|Rings, Modules, and Algebras"
    ]
  ],
  "openalex": "W7212037101",
  "pdf_url": "https://arxiv.org/pdf/2609.06339",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-06",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/8wgc5yxn54vgp.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:19:58Z",
      "native_id": "2609.06339",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.06339"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.RA",
      "subfield": "algebra-number-theory"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Generic Nullity of Generalized Commutators",
  "title_text": "Generic Nullity of Generalized Commutators",
  "type": "preprint",
  "updated_at": "2026-10-30T12:19:58Z",
  "url": "https://research.draconitic.com/paper/8wgc5yxn54vgp",
  "venue": "arXiv",
  "versions": []
}