{
  "abstract": "The discriminants of the derivatives of a real-rooted polynomial are nonnegative polynomials in its roots, and we ask whether this nonnegativity can be certified algebraically in the root differences. We compare three such certificates: sums of squares, positive combinations of symmetrized monomials in squared root differences (the square-graph cone of Alexandersson and Shapiro), and sum-of-squares factorizations of the Hermite matrix of the derivative. We work in the terminal range, where the degree $r=n-k$ of the $k$-th derivative is fixed and the degree $n$ of the polynomial grows. For $r=3$ the discriminant lies in the square-graph cone for every $n$, and we prove the three-graph formula conjectured by Alexandersson and Shapiro. For $r=4$ the two cones separate: the discriminant belongs to the square-graph cone exactly for $4\\le n\\le22$, but it is a sum of squares for every $n\\ge4$. The negative part is certified by an explicit linear functional, the positive part by a rational Gram certificate built from nine orbits of squares under the symmetric group. Thus the square-graph conjecture fails, already at $(n,k)=(23,19)$, while the sum-of-squares question has a positive answer for all $k\\ge n-4$. Finally, we give explicit sum-of-squares factorizations of the Hermite matrices of the terminal cubic and quartic. The quartic factorization yields a uniform proof of the sum-of-squares property for all $n\\ge4$, and we conjecture that such a factorization exists for every derivative; this is verified in the centered outlier limit through terminal degree six.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2605.25743",
  "author_total": 2,
  "authors": [
    {
      "name": "Cordian Riener"
    },
    {
      "name": "Boris Shapiro"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/fcmn67kg66rt9.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2605.25743",
  "fulltext_url": null,
  "id": "fcmn67kg66rt9",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2605.25743",
  "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.CA"
    ],
    [
      "openalex",
      "2607|Advanced Combinatorial Mathematics"
    ]
  ],
  "openalex": "W7162418932",
  "pdf_url": "https://arxiv.org/pdf/2605.25743",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-21",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/fcmn67kg66rt9.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:34:22Z",
      "native_id": "2605.25743",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2605.25743"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CA",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "Discriminants of derivatives: square graphs and sums of squares",
  "title_text": "Discriminants of derivatives: square graphs and sums of squares",
  "type": "preprint",
  "updated_at": "2026-10-30T12:34:22Z",
  "url": "https://research.draconitic.com/paper/fcmn67kg66rt9",
  "venue": "arXiv",
  "versions": []
}