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 Płoski 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.
Citation: Evelia R. GARCÍA BARROSO, Juan Ignacio GARCÍA-GARCÍA, Luis José SANTANA SÁNCHEZ, Alberto VIGNERON-TENORIO. Conductors of Abhyankar-Moh semigroups of even degrees[J]. Electronic Research Archive, 2023, 31(4): 2213-2229. doi: 10.3934/era.2023113
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 Płoski 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.
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Evelia R. GARCÍA BARROSO, Juan Ignacio GARCÍA-GARCÍA, Luis José SANTANA SÁNCHEZ, Alberto VIGNERON-TENORIO. Conductors of Abhyankar-Moh semigroups of even degrees[J]. Electronic Research Archive, 2023, 31(4): 2213-2229. doi: 10.3934/era.2023113
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