Quasi-ordinarization transform of a numerical semigroup

We introduce the quasi-ordinarization transform of a numerical semigroup. This transform will allow to organize all the semigroups of a given genus in a forest rooted at all quasi-ordinary semigroups with the given genus. This construction provides an alternative approach to the conjecture on the increasingness of the number of numerical semigroups for each given genus. We elaborate on the number of nodes at each tree depth in the forest and present a few new conjectures that can be developed in the future. We prove some properties of the quasi-ordinarization transform, its relations with the ordinarization transform, and we also present an alternative approach to the conjecture that the number of numerical semigroups of each given genus is increasing.Comment: arXiv admin note: text overlap with arXiv:1203.500

Modularly equidistant numerical semigroups

The first author was partially supported by MTM-2017-84890-P and by Junta de Andalucia group FQM343. The second author is supported by the project FCT PTDC/MAT/73544/2006).We would like to thank the referees for their comments and suggestions on the manuscript.If S is a numerical semigroup and s E S, we denote by nextS(s) = min (x E S | s < x}. Let a be an integer greater than or equal to two. A numerical semigroup is equidistant modulo a if nextS(s) - s - 1 is a multiple of a for every s E S. In this note, we give algorithms for computing the whole set of equidistant numerical semigroups modulo a with fixed multiplicity, genus, and Frobenius number. Moreover, we will study this kind of semigroups with maximal embedding dimension.Junta de Andalucia MTM-2017-84890-P FQM343 FCT PTDC/MAT/73544/200

The Proportion of Weierstrass Semigroups

We solve a problem of Komeda concerning the proportion of numerical semigroups which do not satisfy Buchweitz' necessary criterion for a semigroup to occur as the Weierstrass semigroup of a point on an algebraic curve. We also show that the family of semigroups known to be Weierstrass semigroups using a result of Eisenbud and Harris, has zero density in the set of all semigroups. In the process, we prove several more general results about the structure of a typical numerical semigroup.Comment: 15 pages. Corrected typos, some minors mathematical changes, added some discussion. To appear in J. Algebr

Isolated factorizations and their applications in simplicial affine semigroups

We introduce the concept of isolated factorizations of an element of a commutative monoid and study its properties. We give several bounds for the number of isolated factorizations of simplicial affine semigroups and numerical semigroups. We also generalize $\alpha$-rectangular numerical semigroups to the context of simplicial affine semigroups and study their isolated factorizations. As a consequence of our results, we characterize those complete intersection simplicial affine semigroups with only one Betti minimal element in several ways. Moreover, we define Betti sorted and Betti divisible simplicial affine semigroups and characterize them in terms of gluings and their minimal presentations. Finally, we determine all the Betti divisible numerical semigroups, which turn out to be those numerical semigroups that are free for any arrangement of their minimal generators

Irreducible numerical semigroups with multiplicity three and four

In this paper we analyze the irreducibility of numerical semigroups with multiplicity up to four. Our approach uses the notion of Kunz-coordinates vector of a numerical semigroup recently introduced in (Blanco-Puerto, 2011). With this tool we also completely describe the whole family of minimal decompositions into irreducible numerical semigroups with the same multiplicity for this set of numerical semigroups. We give detailed examples to show the applicability of the methodology and conditions for the irreducibility of well-known families of numerical semigroups as those that are generated by a generalized arithmetic progression.Comment: 18 page

Cyclotomic numerical semigroups

Given a numerical semigroup $S$, we let $\mathrm P_S(x)=(1-x)\sum_{s\in S}x^s$ be its semigroup polynomial. We study cyclotomic numerical semigroups; these are numerical semigroups $S$ such that $\mathrm P_S(x)$ has all its roots in the unit disc. We conjecture that $S$ is a cyclotomic numerical semigroup if and only if $S$ is a complete intersection numerical semigroup and present some evidence for it. Aside from the notion of cyclotomic numerical semigroup we introduce the notion of cyclotomic exponents and polynomially related numerical semigroups. We derive some properties and give some applications of these new concepts.Comment: 17 pages, accepted for publication in SIAM J. Discrete Mat