Research article

Conductors of Abhyankar-Moh semigroups of even degrees


  • Received: 08 October 2022 Revised: 14 January 2023 Accepted: 07 February 2023 Published: 22 February 2023
  • 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

    Related Papers:

  • 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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    [1] M. Suzuki, Propriétées topologiques des polynômes de deux variables complexes et automorphisms algébriques de l'espace $\mathbb C^2$, J. Math. Soc. Japan, 26 (1974), 241–257.
    [2] S. S. Abhyankar, T. T. Moh, Embeddings of the line in the plane, J. reine angew. Math., 276 (1975), 148–166. https://doi.org/10.1515/crll.1975.276.148 doi: 10.1515/crll.1975.276.148
    [3] H. Bresinsky, Semigroups corresponding to algebroid branches in the plane, Proc. Am. Math. Soc., 32 (1972), 381–384. https://doi.org/10.2307/2037822 doi: 10.2307/2037822
    [4] G. Angermüller, Die Wertehalbgruppe einer ebener irreduziblen algebroiden Kurve, Math. Z., 153 (1977), 267–282. https://doi.org/10.1007/BF01214480 doi: 10.1007/BF01214480
    [5] R. D. Barrolleta, E. R. García Barroso, A. Płoski, On the Abhyankar-Moh inequality, Univ. Iagel. Acta Math., 52 (2015), 7–14.
    [6] E. R. García Barroso, J. Gwoździewicz, A. Płoski, Semigroups corresponding to branches at infinity of coordinate lines in the affine plane, Semigroup Forum, 92 (2016), 534–540. https://doi.org/10.1007/s00233-015-9693-5 doi: 10.1007/s00233-015-9693-5
    [7] J. C. Rosales, P. A. García-Sánchez, Finitely Generated Commutative Monoids, Nova Science Pub Inc, 1999.
    [8] E. R. García Barroso, J. I. García-Garcí, A. Vigneron-Tenorio, Generalized strongly increasing semigroups, Mathematics, 9 (2021), 1370. https://doi.org/10.3390/math9121370 doi: 10.3390/math9121370
    [9] C. Delorme, Sous-monoïdes d'intersection complète de ${\mathbb{N}}$, Ann. Sci. É. Norm. Super., 4 (1976), 145–154.
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