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Conductors of Abhyankar-Moh semigroups of even degrees

In their paper on the embeddings of the line in the plane, Abhyankar and Moh proved an important inequality, now known as the Abhyankar-Moh inequality, which can be stated in terms of the semigroup associated with the branch at infinity of a plane algebraic curve. Barrolleta, García Barroso and P\lo...

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Bibliographic Details
Published in:arXiv.org 2022-09
Main Authors: GarcÍa Barroso, Evelia R, GarcÍa-GarcÍa, Juan Ignacio, Santana SÁnchez, Luis José, Vigneron-Tenorio, Alberto
Format: Article
Language:English
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Summary:In their paper on the embeddings of the line in the plane, Abhyankar and Moh proved an important inequality, now known as the Abhyankar-Moh inequality, which can be stated in terms of the semigroup associated with the branch at infinity of a plane algebraic curve. Barrolleta, García Barroso and P\loski studied the semigroups of integers satisfying the Abhyankar-Moh inequality and call them Abhyankar-Moh semigroups. They described such semigroups with the maximum conductor. In this paper we prove that all possible conductor values are achieved for the Abhyankar-Moh semigroups of even degree. Our proof is constructive, explicitly describing families that achieve a given value as its conductor.
ISSN:2331-8422