Abhyankar-Moh Semigroups for arbitrary hypersurfaces

Fuensanta Aroca; Annel Ayala; Oscar Castañon; Giovanna Ilardi

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arXiv · 2501.16552

Abhyankar-Moh Semigroups for arbitrary hypersurfaces

Fuensanta Aroca·Annel Ayala·Oscar Castañon·Giovanna Ilardi

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Abstract

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 Abhyankar and Moh. 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. Gonz\'alez and Kiyek-Micus, where Gonz\'alez specifically addresses the case of quasi-ordinary singularities, and the extension introduced by Abbas-Assi, can be understood as particular instances within our constructed family.

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Fuensanta Aroca, Annel Ayala, Oscar Castañon, Giovanna Ilardi. 2025-01-27. Abhyankar-Moh Semigroups for arbitrary hypersurfaces. https://arxiv.org/abs/2501.16552

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