{
  "abstract": "In 1959, Snapper showed that the Euler characteristic of the tensor powers of a line bundle on a normal projective scheme is a polynomial, later named the \\emph{Snapper polynomial}. Positivity of coefficients of Snapper polynomials implies various notions of positivity of line bundles, which we study through the lens of magic positivity and real-rootedness. We introduce zonotopal classes in the Grothendieck $K$-ring of vector bundles of the toric variety for any loopless matroid, and prove that their Snapper polynomials are magic positive. Our proof realizes such a Snapper polynomial as a weighted independence polynomial of the Dilworth truncation along certain lines of the matroid. As a consequence, their coefficients are positive, and their $h^{\\ast}$-polynomials are real-rooted. In the realizable case, this polynomial is the multigraded Hilbert polynomial of the wonderful variety embedded in a product of projective lines. We introduce analogous line bundles on the Deligne--Mumford--Knudsen moduli space $\\overline{\\mathcal M}_{0,n}$ and prove that their Snapper polynomials are magic positive. For cotangent line bundles whose first Chern classes are distinct $ψ$-classes, which are not zonotopal, we nonetheless prove that their $h^{\\ast}$-polynomials are real-rooted, whereas their Snapper polynomials are magic positive if and only if $n\\leqslant7$. More generally, we introduce saturated and weakly saturated $K$-classes of matroids, which furnish a sufficient and a necessary condition for magic positivity of Snapper polynomials in terms of their dragon Hall--Rado polymatroids.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.24917",
  "author_total": 1,
  "authors": [
    {
      "name": "Shiyue Li"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/ere1026td299q.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.24917",
  "fulltext_url": null,
  "id": "ere1026td299q",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.24917",
  "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.AG"
    ],
    [
      "arxiv",
      "math.CO"
    ],
    [
      "openalex",
      "2607|Advanced Combinatorial Mathematics"
    ]
  ],
  "openalex": "W7214023206",
  "pdf_url": "https://arxiv.org/pdf/2609.24917",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-21",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/ere1026td299q.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:23:27Z",
      "native_id": "2609.24917",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.24917"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AG",
      "subfield": "algebra-number-theory"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Magic positivity of Snapper polynomials for matroids",
  "title_text": "Magic positivity of Snapper polynomials for matroids",
  "type": "preprint",
  "updated_at": "2026-10-30T12:23:27Z",
  "url": "https://research.draconitic.com/paper/ere1026td299q",
  "venue": "arXiv",
  "versions": []
}