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In this paper, we study a family of binomial ideals defining monomial curves in the n-dimensional affine space determined by n hypersurfaces of the form $x_i^{c_i} - x_1^{u_{i1}} \cdots x_n^{u_{1n}}$ in $\Bbbk [x_1, \ldots , x_n]$ with $u_{ii} = 0, \ i\in \{ 1, \ldots , n\}$ . We prove that the monomial curves in that family are set-theoretic complete intersections. Moreover, if the monomial curve is irreducible, we compute some invariants such as genus, type and Frobenius number of the corresponding numerical semigroup. We also describe a method to produce set-theoretic complete intersection semigroup ideals of arbitrary large height.

Let a,b and n>1 be three positive integers such that a and ∑j=0n−1bj are relatively prime. In this paper, we prove that the toric ideal I associated to the submonoid of N generated by {∑j=0n−1bj}∪{∑j=0n−1bj+a∑j=0i−2bj∣i=2,…,n} is determinantal. Moreover, we prove that for n>3, the ideal I has a unique minimal system of generators if and only if a

In this paper, we introduce and study the numerical semigroups generated by $\{a_1, a_2, \ldots \} \subset \mathbb{N}$ such that $a_1$ is the repunit number in base $b > 1$ of length $n > 1$ and $a_i - a_{i-1} = a\, b^{i-2},$ for every $i \geq 2$, where $a$ is a positive integer relatively prime with $a_1$. These numerical semigroups generalize the repunit numerical semigroups among many others. We show that they have interesting properties such as being homogeneous and Wilf. Moreover, we solve the Frobenius problem for this family, by giving a closed formula for the Frobenius number in terms of $a, b$ and $n$, and compute other usual invariants such as the Ap��ry sets, the genus or the type., 16 pages. An automated PDF production issue has been resolved

In this expository note, we give a self-contained presentation of the equivalence between the opposite category of commutative monoids and that of commutative, monoid $$\Bbbk $$ -schemes that are diagonalizable, for any field $$\Bbbk $$ .

In this paper we study those submonoids of $$\mathbb {N}^d$$ with a nontrivial pseudo-Frobenius set. In the affine case, we prove that they are the affine semigroups whose associated algebra over a field has maximal projective dimension possible. We prove that these semigroups are a natural generalization of numerical semigroups and, consequently, most of their invariants can be generalized. In the last section we introduce a new family of submonoids of $$\mathbb {N}^d$$ and using its pseudo-Frobenius elements we prove that the elements in the family are direct limits of affine semigroups.

In this paper we introduce the notion of extension of a numerical semigroup. We provide a characterization of the numerical semigroups whose extensions are all arithmetic and we give an algorithm for the computation of the whole set of arithmetic extension of a given numerical semigroup. As by-product, new explicit formulas for the Frobenius number and the genus of proportionally modular semigroups are obtained., This article has been accepted for publication in Communications in Algebra, published by Taylor & Francis

Health inequalities are marked in Chile. To address this situation, a health reform was implemented in 2005 that guarantees acute myocardial infarction (AMI) health care for the entire population. We evaluated if the health reform changed AMI early and long-term survival rates by hospital provider (public/private) using a longitudinal population-based study of patients ≥15 years with a first AMI in Chile between 2002 and 2011. Time trends and early (within 28 days) and long-term (29–365 days) survival by age were assessed. We identified 59,557 patients: median age of 64 years; 68.9% men; 83.2% treated at public hospitals; 74.4% with public insurance. Early and long-term case-fatality was higher at public hospitals (14.6% vs 9.3%; P

Producción Científica, This work generalises the short resolution given by Pisón Casares [‘The short resolution of a lattice ideal’, Proc. Amer. Math. Soc. 131(4) (2003), 1081–1091] to any affine semigroup. We give a characterisation of Apéry sets which provides a simple way to compute Apéry sets of affine semigroups and Frobenius numbers of numerical semigroups. We also exhibit a new characterisation of the Cohen–Macaulay property for simplicial affine semigroups., Ministerio de Economía, Industria y Competitividad - Fondo Europeo de Desarrollo Regional (projects MTM2012-36917-C03-01 / MTM2015-65764-C3-1-P)

A simple way of computing the Ap\'ery set of a numerical semigroup (or monoid) with respect to a generator, using Groebner bases, is presented, together with a generalization for affine semigroups. This computation allows us to calculate the type set and, henceforth, to check the Gorenstein condition which characterizes the symmetric numerical subgroups.

We study the structure of the family of numerical semigroups with fixed multiplicity and Frobenius number. We give an algorithmic method to compute all the semigroups in this family. As an application we compute the set of all numerical semigroups with given multiplicity and genus.

Producción Científica, This article deals with the study of the number of limit cycles surrounding a critical point of a quadratic planar vector field, which, in normal form, can be written as x ′ = a1x − y − a3x 2 + (2a2 + a5)xy+a6y 2 , y ′ = x+a1y+a2x 2+(2a3+a4)xy−a2y 2 . In particular, we study the semi-varieties defined in terms of the parameters a1, a2, . . . , a6 where some classical criteria for the associated Abel equation apply. The proofs will combine classical ideas with tools from computational algebraic geometry., Agencia Estatal de Investigación - Fondo Europeo de Desarrollo Regional (grant MTM 2011-22751), Junta de Extremadura (grant GR15055), Ministerio de Economía, Industria y Competitividad - Fondo Europeo de Desarrollo Regional (grant MTM2015-65764-C3-1-P)

Producción Científica, We give two algorithmic procedures to compute the whole set of almost symmetric numerical semigroups with fixed Frobenius number and type, and the whole set of almost symmetric numerical semigroups with fixed Frobenius number. Our algorithms allow to compute the whole set of almost symmetric numerical semigroups with fixed Frobenius number with similar or even higher efficiency that the known ones. They have been implemented in the GAP [The GAP Group, GAP — Groups, Algorithms and Programming, Version 4.8.6; 2016, https://www.gap-system.org] package NumericalSgps [M. Delgado and P. A. García-Sánchez and J. Morais, “numericalsgps”: A GAP package on numerical semigroups, https://github.com/gap-packages/numericalsgps]., Fundação para a Ciência e a Tecnologia (project PTDC/MAT/73544/2006), Ministerio de Economía, Industria y Competitividad - Fondo Europeo de Desarrollo Regional (projects MTM2015-65764-C3-1-P and MTM2014-55367-P)

A congruence on \(\mathbb {N}^n\) is an equivalence relation on \(\mathbb {N}^n\) that is compatible with the additive structure. If \(\Bbbk \) is a field, and I is a binomial ideal in \(\Bbbk [X_1,\dots ,X_n]\) (that is, an ideal generated by polynomials with at most two terms), then I induces a congruence on \(\mathbb {N}^n\) by declaring u and v to be equivalent if there is a linear combination with nonzero coefficients of Xu and Xv that belongs to I. While every congruence on \(\mathbb {N}^n\) arises this way, this is not a one-to-one correspondence, as many binomial ideals may induce the same congruence. Nevertheless, the link between a binomial ideal and its corresponding congruence is strong, and one may think of congruences as the underlying combinatorial structures of binomial ideals. In the current literature, the theories of binomial ideals and congruences on \(\mathbb {N}^n\) are developed separately. The aim of this survey paper is to provide a detailed parallel exposition, that provides algebraic intuition for the combinatorial analysis of congruences. For the elaboration of this survey paper, we followed mainly (Kahle and Miller Algebra Number Theory 8(6):1297–1364, 2014) with an eye on Eisenbud and Sturmfels (Duke Math J 84(1):1–45, 1996) and Ojeda and Piedra Sanchez (J Symbolic Comput 30(4):383–400, 2000).

In this paper, we deal with the problem of uniqueness of minimal system of binomial generators of a semigroup ideal. Concretely, we give different necessary and/or sufficient conditions for uniqueness of such minimal system of generators. These conditions come from the study and combinatorial description of the so-called indispensable binomials in the semigroup ideal., Comment: 11 pages. This paper was initially presented at the II Iberian Mathematical Meeting (http://imm2.unex.es). To appear in the Proc. Amer. Math. Soc

This paper is concerned with the combinatorial description of the graded minimal free resolution of certain monomial algebras which includes toric rings. Concretely, we explicitly describe how the graded minimal free resolution of those algebras is related to the combinatorics of some simplicial complexes. Our description may be interpreted as an algorithmic procedure to partially compute this resolution., Comment: 20 pages, this work was presented as a poster at the 'International Conference on Commutative, Combinatorial and Computational Algebra in Honour of Pilar Pison Casares' Seville, February 11 - 16, 2008. http://departamento.us.es/da/actividades/ciaccc.htm

In Peeva and Sturmfels (1998), the authors introduce a new notion of genericity for lattice ideals; however, the deterministic construction of this kind of “generic lattice ideals” with prescribed properties (such as Betti numbers) is still an open problem that appears to be difficult (cf. Miller and Sturmfels, 2005, Section 9.4). In this note, it is shown that any admissible sequence of Betti numbers for generic lattice ideals of codimension three in a polynomial ring in four variables can occur, by explicitly giving a family of generic lattice ideals, which includes ideals defining toric varieties, in each case.

Let $S_1$ and $S_2$ be two affine semigroups, and let $S$ be the gluing of $S_1$ and $S_2$. Several invariants of $S$ are related to those of $S_1$ and $S_2$; we review some of the most important properties preserved under gluings. The aim of this paper is to prove that this is the case for the Frobenius vector and the Hilbert series. Applications to complete intersection affine semigroups are also given.

We characterize the hull resolution of a monomial curve in three-dimensional affine space, and we compare this resolution with the minimal one. Concretely, we give a necessary and sufficient condition for the minimality of the hull resolution of a monomial curve in three-dimensional affine space in terms of the associated semigroup. We also get a lower bound for the Betti numbers of the minimal free resolution of a generic monomial curve, in four- and five-dimensional affine space.

Using the block vec matrix, I give a necessary and sufficient condition for factorization of a matrix into the Kronecker product of two other matrices. As a consequence, I obtain an elementary algorithmic procedure to decide whether a matrix has a square root for the Kronecker product., 4 pages. This paper has been accepted for publication in The American Mathematical Monthly

Let $\mathbb{N} \mathcal{A}$ be the monoid generated by $\mathcal{A} = {\mathbf{a}_1, ..., \mathbf{a}_n} \subseteq \mathbb{Z}^d.$ We introduce the homogeneous catenary degree of $\mathbb{N} \mathcal{A}$ as the smallest $N \in \mathbb N$ with the following property: for each $\mathbf{a} \in \mathbb{N} \mathcal{A}$ and any two factorizations $\mathbf{u}, \mathbf{v}$ of $\mathbf{a}$, there exists factorizations $\mathbf{u} = \mathbf{w}_1, ..., \mathbf{w}_t = \mathbf{v} $ of $\mathbf{a}$ such that, for every $k, \mathrm{d}(\mathbf{w}_k, \mathbf{w}_{k+1}) \leq N,$ where $\mathrm{d}$ is the usual distance between factorizations, and the length of $\mathbf{w}_k, |\mathbf{w}_k|,$ is less than or equal to $\max{|\mathbf{u}|, |\mathbf{v}|}.$ We prove that the homogeneous catenary degree of $\mathbb{N} \mathcal{A}$ improves the monotone catenary degree as upper bound for the ordinary catenary degree, and we show that it can be effectively computed. We also prove that for half-factorial monoids, the tame degree and the $\omega$-primality coincide, and that all possible catenary degrees of the elements of an affine semigroup of this kind occur as the catenary degree of one of its Betti elements., Comment: 8 pages, 1 figure

In this paper, we prove that every binomial ideal in a polynomial ring over an algebraically closed field of characteristic zero admits a canonical primary decomposition into binomial ideals. Moreover, we prove that this special decomposition is obtained from a cellular decomposition which is also defined in a canonical way and does not depend on the field., Comment: 13 pages, LaTeX. Example 1.5 added. Introduction improved. References removed Typos corrected

In this paper, we study double structures supported on rational normal curves. After recalling the general construction of double structures supported on a smooth curve described in [11], we specialize it to double structures on rational normal curves. To every double structure we associate a triple of integers (2r, g, n) where r is the degree of the support, n ≥ r is the dimension of the projective space containing the double curve, and g is the arithmetic genus of the double curve. We compute also some numerical invariants of the constructed curves, and we show that the family of double structures with a given triple (2r, g, n) is irreducible. Furthermore, we prove that the general double curve in the families associated to (2r, r + 1, r) and (2r, 1, 2r−1) is arithmetically Gorenstein. Finally, we prove that the closure of the locus containing double conics of genus g ≤ −2 form an irreducible component of the corresponding Hilbert scheme, and that the general double conic is a smooth point of that component. Moreover, we prove that the general double conic in ℙ3 of arbitrary genus is a smooth point of the corresponding Hilbert scheme.

A finitely generated commutative monoid is uniquely presented if it has only a minimal presentation. We give necessary and sufficient conditions for finitely generated, combinatorially finite, cancellative, commutative monoids to be uniquely presented. We use the concept of gluing to construct commutative monoids with this property. Finally for some relevant families of numerical semigroups we describe the elements that are uniquely presented., Comment: 13 pages, typos corrected, references updated

After establishing bounds on the Rao function and on the genus of projective curves that generalize the ones in [5] and in [12] , we describe the even G-liaison classes of some unions of curves attaining the bounds, and of more general unions with analogous geometric properties. In particular, we prove that their Hartshorne–Rao module identifies the even G-liaison class.

In this paper, we study the Hilbert scheme of non degenerate locally Cohen- Macaulay projective curves with general hyperplane section spanning a linear space of dimension 2 and minimal Hilbert function. The main result is that those curves are almost always the general element of a generically smooth component Hn,d,g of the corresponding Hilbert scheme. Moreover, we show that the curves with maximal cohomology almost always correspond to smooth points of Hn,d,g.

Eisenbud and Sturmfels’ theoretical study assures that it is possible to find a primary decomposition of binomial ideals into binomial ideals over an algebraically closed field. In this paper we complete the algorithms in Eisenbud and Sturmfels (1996, Duke Math. J., 84, 1–45) by filling in the steps for which the authors say they have not been very precise.

Producción Científica, We establish a one-to-one correspondence between numerical semigroups of genus g and almost symmetric numerical semigroups with Frobenius number F and type F−2g, provided that F is greater than or equal to 4g−1., Ministerio de Economía, Industria y Competitividad - Fondo Europeo de Desarrollo Regional (projects MTM2017-84890-P / MTM2015-65764-C3-1-P)