{
  "abstract": "Let $J$ be the Jordan subalgebra of the free associative algebra $\\Q\\langle X\\rangle$ generated by $X$ and $1$ under $a\\cdot b=\\tfrac12(ab+ba)$. We give a polynomial criterion for an associative polynomial to belong to $J$. For every $n\\ge1$, Robbins' recursion gives a family of operators $T_α\\in\\Q[S_n]$, indexed by compositions of $n$ into parts $1$ and $2$. We prove that right multiplication by $U_n=\\sum_αT_α^\\dagger T_α$ has image $\\SJ(n)$, the space of multilinear Jordan elements. If its characteristic polynomial is $t^{e_n}q_n(t)$, where $q_n(0)\\ne0$, then $Π_n=I-q_n(U_n)/q_n(0)$ is an idempotent and $f\\in\\SJ(n)$ if and only if $fq_n(U_n)=0$, or equivalently $fΠ_n=f$. The same operators give the criterion for arbitrary homogeneous polynomials of degree $n$. We also prove $\\SJ(n)\\cong\\bigoplus_{λ\\vdash n}r_λS^λ$, where $r_λ=\\rankρ_λ(U_n)$. Thus $\\dim\\SJ(n)=\\sum_λd_λr_λ$ and $e_n=\\sum_λd_λ(d_λ-r_λ)$, with $d_λ=\\dim S^λ$. We give a recursion for the corresponding row spaces and use it to compute the multiplicities over $\\Q$ through degree $12$. Exact block characteristic polynomials through degree $7$ and direct computations in small degrees illustrate the criterion.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.08314",
  "author_total": 2,
  "authors": [
    {
      "name": "F. Mashurov"
    },
    {
      "name": "B. Sartayev"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/4vsz9g9d7xrn0.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.08314",
  "fulltext_url": null,
  "id": "4vsz9g9d7xrn0",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.08314",
  "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",
      "2602|Advanced Topics in Algebra"
    ]
  ],
  "openalex": "W7211995826",
  "pdf_url": "https://arxiv.org/pdf/2609.08314",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-20",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/4vsz9g9d7xrn0.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:34:08Z",
      "native_id": "2609.08314",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.08314"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.RA",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "A polynomial criterion for Jordan elements in a free associative algebra",
  "title_text": "A polynomial criterion for Jordan elements in a free associative algebra",
  "type": "preprint",
  "updated_at": "2026-10-30T12:34:08Z",
  "url": "https://research.draconitic.com/paper/4vsz9g9d7xrn0",
  "venue": "arXiv",
  "versions": []
}