{
  "abstract": "An observation of J-P.~Serre implies that the generic monic cubic polynomial, unique among generic monic polynomials of degree at least two, has a root that is a power series in the discriminant; Serre asked for a formula. We give formulas that work over any field with an absolute value and in every characteristic. Over a complete non-archimedean field we identify the root intrinsically: it is the isolated root, the one farthest from the others. We also answer the next case of Serre's question, computing explicitly the distinguished ramified quadratic factor of the generic monic quartic. The methods combine Hensel's lemma, identities for generalized binomial series, and elementary non-archimedean analysis.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2605.25992",
  "author_total": 3,
  "authors": [
    {
      "name": "Jason Bland"
    },
    {
      "name": "Skip Garibaldi"
    },
    {
      "name": "Joel Rosenberg"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/e3gw27bw054yb.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2605.25992",
  "fulltext_url": null,
  "id": "e3gw27bw054yb",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2605.25992",
  "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.RA"
    ],
    [
      "openalex",
      "2608|Algebraic Geometry and Number Theory"
    ]
  ],
  "openalex": "W7162467448",
  "pdf_url": "https://arxiv.org/pdf/2605.25992",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-20",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/e3gw27bw054yb.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:34:08Z",
      "native_id": "2605.25992",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2605.25992"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.RA",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "Root of the generic cubic as a power series in the discriminant",
  "title_text": "Root of the generic cubic as a power series in the discriminant",
  "type": "preprint",
  "updated_at": "2026-10-30T12:34:08Z",
  "url": "https://research.draconitic.com/paper/e3gw27bw054yb",
  "venue": "arXiv",
  "versions": []
}