An Overview of the Computational Aspects of Nonunique Factorization Invariants

García-Sánchez, Pedro A.

doi:10.1007/978-3-319-38855-7_7

Cited by 25 publications

(27 citation statements)

References 37 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…The study of these simplicial affine semigroups may be an interesting topic for further research. Thirdly, it is possible that some non-unique factorization invariants ( [17,14] and [2, Chapter 5]) may be computed or bounded for Betti divisible semigroups. Again this topic is not related with isolated factorizations and, therefore, we did not dig into this topic in the present work.…”

Section: Further Researchmentioning

confidence: 99%

See 1 more Smart Citation

Isolated factorizations and their applications in simplicial affine semigroups

García-Sánchez

Herrera-Poyatos

2019

J. Algebra Appl.

View full text Add to dashboard Cite

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 α-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.

show abstract

Section: Further Researchmentioning

confidence: 99%

“…The set of the Betti elements of M is denoted by Betti(M ). Betti elements and their R-classes can be used to characterize the minimal presentations of M , see Section 2.5, and, thus, they are widely studied in the theory of commutative monoids (see for instance [15,14]…”

Section: Introductionmentioning

confidence: 99%

Isolated factorizations and their applications in simplicial affine semigroups

García-Sánchez

Herrera-Poyatos

2019

J. Algebra Appl.

View full text Add to dashboard Cite

show abstract

“…Such minimal presentations arise in the study of toric ideals, where they correspond to minimal generating sets for kernels of monomial maps [7], and algebraic statistics, where they correspond to Markov bases [9]. Additionally, many arithmetic invariants of interest in combinatorial commutative algebra and factorization theory can be easily recovered (both theoretically and computationally) from a minimal presentation, making them a particularly useful tool in computational algebra [10,14].…”

Section: Introductionmentioning

confidence: 99%

Minimal presentations of shifted numerical monoids

Conaway

Gotti

Horton

et al. 2018

Int. J. Algebra Comput.

View full text Add to dashboard Cite

A numerical monoid is an additive submonoid of the non-negative integers. Given a numerical monoid S, consider the family of "shifted" monoids M n obtained by adding n to each generator of S. In this paper, we examine minimal relations among the generators of M n when n is sufficiently large, culminating in a description that is periodic in the shift parameter n. We explore several applications to computation, combinatorial commutative algebra, and factorization theory.

show abstract

“…Our intention is to provide as many functions as possible for affine semigroups as we offer for numerical semigroups. Along this line, we present methods for membership, computing minimal presentations, determine gluings, Betti and primitive elements, and the whole series of procedures for nonunique factorization invariants (an overview of the existing methods for the calculation of these invariants can be found in [30]). New procedures are now under development based in Hilbert functions and binomial ideals ( [37]).…”

Section: 4mentioning

confidence: 99%