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<p>For an arbitrary hypersurface singularity, we construct a family of semigroups associated with algebraically closed fields that arise as an infinite union of rings of series. These semigroups extend the value semigroup of a plane curve studied by S. S. Abhyankar and T. T. Moh [<CitationRef AdditionalCitationIDS="CR3" CitationID="CR2">2</CitationRef>–<CitationRef CitationID="CR4">4</CitationRef>]. The algebraically closed fields under consideration possess a natural valuation that induces a corresponding value semigroup. We establish the necessary conditions under which these semigroups are independent of the choice of the root. Moreover, the extensions proposed by P. D. González, K.-H. Kiyek and M. Micus [<CitationRef CitationID="CR15">15</CitationRef>, <CitationRef CitationID="CR17">17</CitationRef>], where they specifically address the case of quasi-ordinary singularities, and the extensions introduced by A. Sathaye [<CitationRef CitationID="CR25">25</CitationRef>] and by A. Ali and A. Assi [<CitationRef CitationID="CR5">5</CitationRef>], can be understood as particular instances within our constructed family.</p>

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Abhyankar–Moh semigroups for arbitrary hypersurfaces

  • 期刊论文

  • 发表日期：

    2026年7月29日

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  • Fuensanta Aroca,
  • Annel Ayala,
  • Oscar Castañon,
  • Giovanna Ilardi

摘要

For an arbitrary hypersurface singularity, we construct a family of semigroups associated with algebraically closed fields that arise as an infinite union of rings of series. These semigroups extend the value semigroup of a plane curve studied by S. S. Abhyankar and T. T. Moh [2–4]. The algebraically closed fields under consideration possess a natural valuation that induces a corresponding value semigroup. We establish the necessary conditions under which these semigroups are independent of the choice of the root. Moreover, the extensions proposed by P. D. González, K.-H. Kiyek and M. Micus [15, 17], where they specifically address the case of quasi-ordinary singularities, and the extensions introduced by A. Sathaye [25] and by A. Ali and A. Assi [5], can be understood as particular instances within our constructed family.