{
  "abstract": "Let $R$ be a $d$-dimensional Noetherian local ring with maximal ideal $m_R$ and let $I$ be an ideal of $R$. The epsilon multiplicity $ε(I)$ of $I$ is defined by Ulrich and Validashti in \\cite{UV} as a limsup of normalized lengths. We show in this paper that the epsilon multiplicity is always a limit of these normalized lengths. Specifically, $$ ε(I)= \\lim_{n\\rightarrow\\infty} \\frac{\\ell_R((I^n)^{\\rm sat}/I^n)}{n^d/d!}. $$ We prove that the epsilon multiplicity satisfies an associativity formula, analogous to that of the ordinary multiplicity. We prove these results by comparing the epsilon multiplicity to a suitable limsup of normalized Amao multiplicities, the Amao epsilon multiplicity $aε(I)$. We show that the epsilon multiplicity is always equal to the Amao epsilon multiplicity, which is itself a limit.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2608.12238",
  "author_total": 1,
  "authors": [
    {
      "name": "Steven Dale Cutkosky"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/9a47gjh5repfw.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2608.12238",
  "fulltext_url": null,
  "id": "9a47gjh5repfw",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2608.12238",
  "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.AC"
    ],
    [
      "arxiv",
      "math.AG"
    ],
    [
      "openalex",
      "2602|Commutative Algebra and Its Applications"
    ]
  ],
  "openalex": "W7202354453",
  "pdf_url": "https://arxiv.org/pdf/2608.12238",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-04",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/9a47gjh5repfw.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:27:10Z",
      "native_id": "2608.12238",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2608.12238"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AC",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "Limits of Amao multiplicities and Epsilon multiplicity",
  "title_text": "Limits of Amao multiplicities and Epsilon multiplicity",
  "type": "preprint",
  "updated_at": "2026-10-30T12:27:10Z",
  "url": "https://research.draconitic.com/paper/9a47gjh5repfw",
  "venue": "arXiv",
  "versions": []
}