{
  "abstract": "For finite-dimensional complex power-associative algebras $\\mathcal{A}$ and $\\mathcal{B}$, each equipped with its unique Hausdorff vector topology, we characterize continuous spectrum-shrinking maps $ϕ:\\mathcal{A}\\to\\mathcal{B}$, i.e. maps satisfying $\\mathrm{sp}_{\\mathcal{B}}(ϕ(x)) \\subseteq\\mathrm{sp}_{\\mathcal{A}}(x)$ for all $x\\in\\mathcal{A}$. The spectrum of an element is the zero set of its minimal polynomial, computed in the unitization for nonunital algebras. No such map exists when $\\mathcal{A}$ is unital and $\\mathcal{B}$ is nonunital. Otherwise, we symmetrize both products by setting $x\\circ y:=\\tfrac{1}{2}(xy+yx)$ and factor out the nilradicals to obtain semisimple Jordan quotients. We prove that a continuous map $ϕ$ is spectrum-shrinking precisely when, for every $x\\in\\mathcal{A}$, Jacobson's generic minimal polynomials of the simple quotient coordinates of $ϕ(x)$ are products of fixed nonnegative integer powers of those of $x$, with an additional fixed monomial factor allowed when $\\mathcal{A}$ is nonunital. Comparing degrees gives linear Diophantine equations characterizing existence, and every nonnegative integer solution is realized by a polynomial spectrum-shrinking map. Spectrum preservation, meaning equality of spectra, is equivalent to additional positivity conditions. We also determine when every continuous spectrum-shrinking map preserves spectra and show that every continuous map $f:\\mathcal{A}\\to\\mathbb{C}$ with $f(x)\\in\\mathrm{sp}_{\\mathcal{A}}(x)$ for all $x\\in\\mathcal{A}$ is $\\mathbb{C}$-linear and satisfies $f(x\\circ y)=f(x)f(y)$ for all $x,y\\in\\mathcal{A}$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.29393",
  "author_total": 1,
  "authors": [
    {
      "name": "Ilja Gogić"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/az5rh3jxb6k7d.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.29393",
  "fulltext_url": null,
  "id": "az5rh3jxb6k7d",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.29393",
  "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",
      "2610|Advanced Operator Algebra Research"
    ]
  ],
  "openalex": "W7223221180",
  "pdf_url": "https://arxiv.org/pdf/2610.29393",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-21",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/az5rh3jxb6k7d.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:34:22Z",
      "native_id": "2610.29393",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.29393"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.RA",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "Continuous spectrum-shrinking maps between finite-dimensional power-associative algebras",
  "title_text": "Continuous spectrum-shrinking maps between finite-dimensional power-associative algebras",
  "type": "preprint",
  "updated_at": "2026-10-30T12:34:22Z",
  "url": "https://research.draconitic.com/paper/az5rh3jxb6k7d",
  "venue": "arXiv",
  "versions": []
}