Evora, 18-20 July 2023

Manuel Delgado, Universidade do Porto , Still counting numerical semigroups

Abstract

There are two kinds of results related to counting numerical semigroups:

experimental results, usually relying on some deep computational and algorithmic work;

asymptotic results, generally depending upon combinatorial and statistical methods.

There are many results involving the following two distinct ways of counting numerical semigroups:

counting by Frobenius number (representing by N(f) the number of numerical semigroups of Frobenius number f, what is N(f) for some values of f, and what is the order of magnitude of N(f) when f tends to infinity?);

counting by genus (representing by n(g) the number of numerical semigroups with genus g, what is n(g) for some values of g, and what can be said about the behaviour of the sequence (n(g))?).

I will give an overview of the existing results, some of which are recent.

Partially supported by Proyecto “Monoides y semigrupos afines (ProyExcel_00868)“, Proyecto financiado en la convocatoria 2021 de Ayudas a Proyectos de Excelencia, en régimen de concurrencia competitiva, destinadas a entidades calificadas como Agentes del Sistema Andaluz del Conocimiento, en el ámbito del Plan Andaluz de Investigación, Desarrollo e Innovación (PAIDI 2020). Consejería de Universidad, Investigación e Innovación de la Junta de Andalucía.

 

Evelia R. García Barroso, Universidad de La Laguna, Conductors of Abhyankar-Moh semigroups of even degrees

Abstract

In their paper [1] 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. In [2] were studied the semigroups of integers satisfying the Abhyankar-Moh inequality and call them Abhyankar-Moh semigroups and described such semigroups with the maximum conductor. In this talk I will show that all possible conductor values are achieved for the Abhyankar-Moh semigroups of even degree. The proof is constructive, explicitly describing families that achieve a given value as its conductor.

This talk is based on the results of [3].

Partially supported by “Monoides afines y sus aplicaciones (PR2022-011)”, Proyecto de investigación del Plan Propio – UCA 2022-2023, Universidad de Cádiz, and by the grant PID2019-105896GB-I00 funded by MCIN/AEI/10.13039/501100011033.

References

[1] S. S. Abhyankar, T. T. Moh. Embeddings of the line in the plane, J. reine angew. Math., 276 (1975), 148Ð166. https://doi.org/10.1515/crll.1975.276.148.

[2] R. Barrolleta, E.R. García Barroso, A. P\l oski. On the Abhyankar-Moh inequality, Universitatis Iagellonicae Acta Mathematica, LII (2015), 7-14. doi:10.4467/20843828AM.15.001.3727.

[3] E.R. García Barroso, J.I. García-García, L.J. Santana Sánchez, A. Vigneron-Tenorio. Conductors of Abhyankar-Moh semigroups of even degrees, Electronic Research Archive, 31(4): 2213Ð2229. DOI: 10.3934/era.2023113

 

Isaac Guzmán-Maya, Universidad de Cádiz, Walnut: Introduction and applications

Abstract

Walnut software basics and operation. Working with automatic sequences: the evil numbers, the odious numbers, and the lower and upper Wythoff sequences.

Supported by “Monoides afines y sus aplicaciones (PR2022-011)”, Proyecto de investigación del Plan Propio – UCA 2022-2023, Universidad de Cádiz.

 

David Llena, Universidad de Almería, Betti elements on affine semigroups

Abstract

Let an ideal of , that is, , where is a finite set. Consider    which is a submonoid of (could be non finitely generated). However, if it is imposed that   has a finite number of elements, or equivalently has one (or more) element in every axis, we have a generalized numerical semigroup. The set of minimal generators of these semgiroups can be obtained from the set of gaps , but, en general, it is a very large set. In this work in progress, we try to give a characterization of the set of Betti elements, whitout using the set of minimal generators.

Partially supported by Proyecto “Monoides y semigrupos afines (ProyExcel_00868)“, Proyecto financiado en la convocatoria 2021 de Ayudas a Proyectos de Excelencia, en régimen de concurrencia competitiva, destinadas a entidades calificadas como Agentes del Sistema Andaluz del Conocimiento, en el ámbito del Plan Andaluz de Investigación, Desarrollo e Innovación (PAIDI 2020). Consejería de Universidad, Investigación e Innovación de la Junta de Andalucía.

 

Daniel Marín-Aragón, Universidad de Cádiz, A family of non-Weierstrass semigroups.

Abstract

The -semigroups are a family of numerical semigroups defined by the position of their pseudo-Frobenius numbers in their sets of gaps. We study this family and show that they can be written as the interesection of Weierstrass semigroups but they are non-Weierstrass.

Supported by “Monoides afines y sus aplicaciones (PR2022-011)”, Proyecto de investigación del Plan Propio – UCA 2022-2023, Universidad de Cádiz.

 

Carlos-Jesús Moreno-Ávila, Universitat Jaume I, Some combinatorial thoughts around semigroups at infinity

Abstract

Let be a curve on with only one place at infinity at a point , and let be its semigroup at infinity, i.e. the additive submonoid of consisting of the orders —with negative sign— of the poles of the regular functions around (but not in)  After a theorem by Abhyankar and Moh, we can associate to a so-called -sequence in which is a system of generators of , by no means unique.

Curves with only one place at infinity are relevant, for instance they play an important role in the study of the Jacobian conjecture. However, not so much is known about  from a combinatorial point of view. In this talk we review the previous concepts and results and we see some properties of the -sequences; in particular, we introduce the notion of minimal -sequence as that generated only by the minimal elements of the semigroup at infinity. In addition, we show how we can compute the remaining -sequences associating with the same semigroup at infinity. Finally, we see algorithm procedures to compute minimal -sequences.

This talk is based on a joint work in progress with C. Galindo, F. Monserrat and J.-J. Moyano-Fernández.

 

María Ángeles Moreno-Frías, Universidad de Cádiz, Frobenius pseudo-variety of numerical semigroups with a given multiplicity and ratio

Abstract

Let be a numerical semigroup. Denote by , and the minimal system of generators, the muliplicity and the ratio of respectively.

In this work, we study the set of all numerical semigroups with multiplicity and ratio , denoted by . In particular, we prove that is a Frobenius pseudo-variety (see [2]) and we present some algorithms which compute all the elements of with a given genus or with a given Frobenius number.

Let . The results of [3] allow us to state that an element of is determined by a subset of . As a consequence, we give formulas to obtain the Frobenius number and the genus of an element of from its subset of  associated.

This talk is based on a work which has been recently published in Appl. Algebr. Eng. Commun. Comput. (see [1]).

Supported by “Monoides afines y sus aplicaciones (PR2022-011)”, Proyecto de investigación del Plan Propio – UCA 2022-2023, Universidad de Cádiz.

References

[1] M.A. Moreno-Frías and J. C. Rosales. Frobenius pseudo-varieties of numerical semigroups with a given multiplicity and ratio, Appl. Alg. Eng. Comm. Comp. https://doi.org/10.1007/s00200-023-00610-w.

[2] A. M. Robles-Pérez and J. C. Rosales. Frobenius pseudo-varieties in numerical semigroups, Ann. Mat. Pura Appl. 194 (2015), 275–287.

[3] J. C. Rosales and M. B. Branco. The Frobenius problem for numerical semigorups, J. Number Theory 131 (2011), 2310–2319.

 

Ignacio Ojeda, Universidad de Extremadura, On the depth of simplicial affine semigroup algebras

Abstract

In this talk, we recall the different characterizations of the depth of a monomial algebra. We conjecture that the property of a monomial algebra of having extreme depth is determined by the existence of maximal elements in certain Avery sets when the monomial structure comes from simplicial affine semigroups, exhibiting relevant evidences in low-dimensional cases.

Supported by “Monoides afines y sus aplicaciones (PR2022-011)”, Proyecto de investigación del Plan Propio – UCA 2022-2023, Universidad de Cádiz.

 

Adrián Sánchez-Loureiro, Universidad de Cádiz, On affine C-semigroups

Abstract

Let be an integer cone. An affine semigroup is called -semigroup whenever is finite. We show the generalization of several properties of numerical semigroups to -semigroups.

Joint work with J. I. García-García, D. Marín-Aragón and A. Vigneron-Tenorio.

Supported by “Monoides afines y sus aplicaciones (PR2022-011)”, Proyecto de investigación del Plan Propio – UCA 2022-2023, Universidad de Cádiz.

Reference

García-García, J. I.; Marín-Aragón, D.; Sánchez-Loureiro, A.; Vigneron-Tenorio, A. Some properties of affine -semigroups. arXiv:2305.05044

 

José Luis Santana Sánchez, Universidad de Valladolid, The p-Frobenius vector of affine semigroups

Abstract

Given a numerical semigroup , minimally generated by , we denote by . In this notation, the Frobenius number of is the maximum integer such that . A natural generalization of the Frobenius number presented in the literature is the so called -Frobenius number, which is the maximum integer such that .

In this talk we consider a further generalization, and we define the -Frobenius vector of an affine semigroup with a monomial order on  in an analogous way. Our goal is to present some algorithms that compute the -Frobenius vectors as well as characterizing how these vectors behave when considering gluing of affine semigroups.

 

Márcio André Traesel, Instituto Federal de São Paulo, Frobenius R-variety in numerical semigroups containing a given semigroup

Abstract

Let be a numerical semigroup and . We prove that is Frobenius R-variety that can be arranged in a tree rooted in . We introduce the concepts of Frobenius and genus number of restricted to (respectively and ). We give formulas for , and generalizations of the Amorós’s and Wilf’s conjecture. Moreover, we will show that most of the results of irreducibility can be generalized to -irreducibility.