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  "title": "An eight-term bound for degree-twenty Casas–Alvero polynomials",
  "shortTitle": "Seven terms are not enough",
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  "oneLine": "A candidate sparsity theorem excludes every degree-twenty counterexample with seven or fewer terms in centered form.",
  "abstract": "This unrefereed candidate proves an eight-term lower bound for nontrivial characteristic-zero Casas–Alvero polynomials of degree twenty after centering at the nineteenth-derivative root. Fourteen exact seven-term supports and nineteen compatible residue cases are excluded, in addition to the lower-term theorem. The local arguments allow arbitrary ramification and coefficients that reduce to zero. The result does not settle degree twenty or the full conjecture.",
  "datePublished": "2026-09-24",
  "dateModified": "2026-09-24",
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    "Anonymous"
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    "name": "Casas–Alvero conjecture: degree-twenty sparsity restriction",
    "url": "https://doi.org/10.1090/S0025-5718-2014-02809-3"
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  "keywords": [
    "Casas-Alvero conjecture",
    "sparse polynomials",
    "Hasse derivatives",
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  "keyResults": [
    "A centered nontrivial degree-twenty Casas–Alvero polynomial requires at least eight nonzero monomials, counting its leading term.",
    "All fourteen seven-term supports and nineteen compatible seed cases are excluded.",
    "A separated-cluster lemma turns the residue-field quartic property into an exact root collision under explicit witness hypotheses."
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  "openProblems": [
    "Obtain unaffiliated specialist examination of the full valuation argument and independent reproduction of its finite dependencies.",
    "Determine which additional degree-twenty strata can be excluded while allowing arbitrary ramification.",
    "Test further applications of the separated-cluster lemma with every derivative-witness hypothesis retained."
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  "relatedWorks": [
    {
      "citation": "Castryck, Laterveer and Ounaïes (2014): mean-root simplicity, determinant restrictions and degree-twelve computation.",
      "url": "https://doi.org/10.1090/S0025-5718-2014-02809-3",
      "doi": "10.1090/S0025-5718-2014-02809-3"
    },
    {
      "citation": "De Frutos Marín (2013): arithmetic and sparse-support criteria.",
      "url": "https://doi.org/10.35376/10324/3602",
      "doi": "10.35376/10324/3602"
    },
    {
      "citation": "Massri (v6, 2023): degree twenty with three recycled roots, a different invariant.",
      "url": "https://arxiv.org/abs/1806.09561v6",
      "doi": "10.48550/arXiv.1806.09561"
    },
    {
      "citation": "Marashdeh (2026): descent-set and support obstructions.",
      "url": "https://arxiv.org/abs/2608.14726v1",
      "doi": "10.48550/arXiv.2608.14726"
    },
    {
      "citation": "Ghosh (v2, March 2026): unrefereed all-degree proof claim, not a premise of this paper.",
      "url": "https://arxiv.org/abs/2501.09272v2",
      "doi": "10.48550/arXiv.2501.09272"
    }
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