{
  "abstract": "This paper introduces and studies $(2,3)$-palintropic algebras, a class of commutative algebras defined by the identity $(x^{2})^3 - (x^{3})^2 = 0$. This specific relation is the simplest generator of the $2$-dimensional space of minimal-degree evanescent identities in degree $6$, and encompasses several well-studied structures, including Jordan and medial algebras. The primary motivation for investigating these algebras lies in their trivial Peirce polynomials, which removes a priori restrictions on the spectrum of the multiplication operator associated with an idempotent. This paper reviews and further develops the theory of Peirce operators, Peirce polynomials, and second-order linearizations. It shows that despite the triviality of the Peirce polynomial, any idempotent $c$ admits well-behaved, explicit fusion rules for multiplication between its $λ$-Peirce spaces for $λ\\neq \\tfrac{1}{2}$. Furthermore, multiplication by a nonsingular idempotent is an algebra homomorphism. Finally, there are given concrete examples of $(2,3)$-palintropic algebras, together with applications of these algebraic structures to commutative polynomial maps.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2606.27496",
  "author_total": 2,
  "authors": [
    {
      "name": "Daniel J. F. Fox"
    },
    {
      "name": "Vladimir G. Tkachev"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/9mestq48h610b.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2606.27496",
  "fulltext_url": null,
  "id": "9mestq48h610b",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2606.27496",
  "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.RA"
    ],
    [
      "openalex",
      "2602|Advanced Topics in Algebra"
    ]
  ],
  "openalex": "W7166506958",
  "pdf_url": "https://arxiv.org/pdf/2606.27496",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-21",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/9mestq48h610b.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:34:22Z",
      "native_id": "2606.27496",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2606.27496"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.RA",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "Commutative algebras satisfying univariate identities with vanishing Peirce polynomial",
  "title_text": "Commutative algebras satisfying univariate identities with vanishing Peirce polynomial",
  "type": "preprint",
  "updated_at": "2026-10-30T12:34:22Z",
  "url": "https://research.draconitic.com/paper/9mestq48h610b",
  "venue": "arXiv",
  "versions": []
}