{
  "abstract": "The $\\mathbb{A}^1$-Milnor number of a polynomial map is a quadratic enrichment of the classical Milnor number, defined as the local $\\mathbb{A}^1$-Brouwer degree of the gradient. We study which elements of the Grothendieck-Witt ring of nondegenerate symmetric bilinear forms arise as $\\mathbb{A}^1$-Milnor numbers. Over fields of characteristic not $2$, $3$, or $5$, we show that every element of the Grothendieck-Witt ring of rank at most $7$ that is a local $\\mathbb{A}^1$-Brouwer degree is also an $\\mathbb{A}^1$-Milnor number. Unlike for local $\\mathbb{A}^1$-Brouwer degrees, however, in general not all $\\mathbb{A}^1$-Milnor numbers can be obtained from two-variable polynomials. Over an arbitrary field of characteristic not $2$, $3$, $5$, or $7$, we characterize the elements of the Grothendieck-Witt ring of rank at most $8$ that are $\\mathbb{A}^1$-Milnor numbers of two-variable polynomials. Over subfields of $\\mathbb{R}$, this implies that there are local $\\mathbb{A}^1$-Brouwer degrees of ranks $4$, $6$, and $8$ that are not the $\\mathbb{A}^1$-Milnor number of a two-variable polynomial. We also prove that under a mild hypothesis one can reduce the number of variables of a polynomial while preserving the $\\mathbb{A}^1$-Milnor number up to multiplication by a unit.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.14506",
  "author_total": 4,
  "authors": [
    {
      "name": "C-Y. Jean Chan"
    },
    {
      "name": "Thomas Hagedorn"
    },
    {
      "name": "Joel Louwsma"
    },
    {
      "name": "Michael Wijaya"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/1pg42mtcwx7hm.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.14506",
  "fulltext_url": null,
  "id": "1pg42mtcwx7hm",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.14506",
  "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.AT"
    ],
    [
      "openalex",
      "2608|Algebraic Geometry and Number Theory"
    ]
  ],
  "openalex": "W7221780656",
  "pdf_url": "https://arxiv.org/pdf/2610.14506",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-10",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/1pg42mtcwx7hm.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:29:15Z",
      "native_id": "2610.14506",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.14506"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AG",
      "subfield": "algebra-number-theory"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.AT",
      "subfield": "geometry-topology"
    }
  ],
  "title": "Characterization of $\\mathbb{A}^1$-Milnor numbers of small rank",
  "title_text": "Characterization of 𝔸^1-Milnor numbers of small rank",
  "type": "preprint",
  "updated_at": "2026-10-30T12:29:15Z",
  "url": "https://research.draconitic.com/paper/1pg42mtcwx7hm",
  "venue": "arXiv",
  "versions": []
}