Complete intersections in certain affine and projective monomial curves

Let $k$ be an arbitrary field, the purpose of this work is to provide families of positive integers $\mathcal{A} = \{d_1,\ldots,d_n\}$ such that either the toric ideal $I_{\mathcal A}$ of the affine monomial curve $\mathcal C = \{(t^{d_1},\ldots,\,t^{d_n}) \ | \ t \in k\} \subset \mathbb{A}_k^n$ or the toric ideal $I_{\mathcal A^{\star}}$ of its projective closure ${\mathcal C^{\star}} \subset \mathbb{P}_k^n$ is a complete intersection. More precisely, we characterize the complete intersection property for $I_{\mathcal A}$ and for $I_{\mathcal A^{\star}}$ when: (a) $\mathcal{A}$ is a generalized arithmetic sequence, (b) $\mathcal{A} \setminus \{d_n\}$ is a generalized arithmetic sequence and $d_n \in \mathbb{Z}^+$, (c) $\mathcal{A}$ consists of certain terms of the $(p,q)$-Fibonacci sequence, and (d) $\mathcal{A}$ consists of certain terms of the $(p,q)$-Lucas sequence. The results in this paper arise as consequences of those in Bermejo et al. [J. Symb. Comput. 42 (2007)], Bermejo and Garc\'{\i}a-Marco [J. Symb. Comput. (2014), to appear] and some new results regarding the toric ideal of the curve.Comment: 22 pages. To appear in Bulletin of the Brazilian Mathematical Societ

New spaces of matrices with operator entries

In this paper, we will consider matrices with entries in the space of operators $\mathcal{B}(H)$, where $H$ is a separable Hilbert space and consider the class of matrices that can be approached in the operator norm by matrices with a finite number of diagonals. We will use the Schur product with Toeplitz matrices generated by summability kernels to describe such a class and show that in the case of Toeplitz matrices it can be identified with the space of continuous functions with values in $\mathcal B(H)$. We shall also introduce matriceal versions with operator entries of classical spaces of holomorphic functions such as $H^\infty(\mathbb{D})$ and $A(\mathbb{D})$ when dealing with upper triangular matrices

A class of Schur multipliers of matrices with operator entries

In this paper, we will consider matrices with entries in the space of operators $\mathcal{B}(H)$, where $H$ is a separable Hilbert space, and consider the class of (left or right) Schur multipliers that can be approached in the multiplier norm by matrices with a finite number of diagonals. We will concentrate on the case of Toeplitz matrices and of upper triangular matrices to get some connections with spaces of vector-valued functions.Comment: arXiv admin note: text overlap with arXiv:1810.0781

The cyclicity of polynomial centers via the reduced bautin depth

We describe a method for bounding the cyclicity of the class of monodromic singularities of polyn omial planar families of vector fields X λ with an analytic Poincar e first return map having a polynomial Bautin ideal B in the ring of polynomials in the parameters λ of the family. This class includes the nondegenerate centers, generic nilpotent centers and also some degenerate centers. This method can work even in the case in which B is not radical by studying the stabilization of the integral closures of an ascending chain of polynomial ideals that stabilizes at B. The approach is based on computational algebra methods for determining a minimal basis of the integral closurē B of B. As far as we know, the obtained cyclicity bound is the minimum found in the literature.The first author was partially supported by a MINECO grant number MTM2014-53703-P and by a CIRIT grant number 2014 SGR 1204

Integrable zero-Hopf singularities and 3-dimensional centers

In this paper we show that the well-known Poincaré-Lyapunov nondegenerate analytic center problem in the plane and its higher dimensional version expressed as the 3-dimensional center problem at the zero-Hopf singularity have a lot of common properties. In both cases the existence of a neighborhood of the singularity in the phase space completely foliated by periodic orbits (including equilibria) is characterized by the fact that the system is analytically completely integrable. Hence its Poincaré-Dulac normal form is analytically orbitally linearizable. There also exists an analytic Poincar\'e return map and, when the system is polynomial and parametrized by its coefficients, the set of systems with centers corresponds to an affine variety in the parameter space of coefficients. Some quadratic polynomial families are considered.The author is partially supported by a MINECO grant number MTM2014-53703-P and by a CIRIT grant number 2014 SGR 1204

On divisor-closed submonoids and minimal distances in finitely generated monoids

We study the lattice of divisor-closed submonoids of finitely generated cancellative commutative monoids. In case the monoid is an affine semigroup, we give a geometrical characterization of such submonoids in terms of its cone. Finally, we use our results to give an algorithm for computing $\Delta^*(H)$ the set of minimal distance of $H$

Proportionally modular affine semigroups

This work introduces a new kind of semigroup of $\N^p$ called proportionally modular affine semigroup. These semigroups are defined by modular Diophantine inequalities and they are a generalization of proportionally modular numerical semigroups. We prove they are finitely generated and we give an algorithm to compute their minimal generating sets. We also specialise on the case $p=2$. For this case, we provide a faster algorithm to compute their minimal system of generators and we prove they are Cohen-Macaulay and Buchsbaum. Besides, the Gorenstein property is charactized, and their (minimal) Fr\"obenius vectors are determinated

Computation of the w-primality and asymptotic w-primality with applications to numerical semigroups

We give an algorithm to compute the $\omega$-primality of finitely generated atomic monoids. Asymptotic \w-primality is also studied and a formula to obtain it in finitely generated quasi-Archimedean monoids is proven. The formulation is applied to numerical semigroups, obtaining an expression of this invariant in terms of its system of generators

Affine convex body semigroups and Buchsbaum rings

In this work, new families of Buchsbaum rings, developed by mean of convex body semigroups, are presented. We characterize Buchsbaum circle and convex polygonal semigroups and describe algorithmic methods to check these characterizations

Computation of Delta sets of numerical monoids

Let $\{a_1,\dots,a_p\}$ be the minimal generating set of a numerical monoid $S$. For any $s\in S$, its Delta set is defined by $\Delta(s)=\{l_{i}-l_{i-1}|i=2,\dots,k\}$ where $\{l_1<\dots<l_k\}$ is the set $\{\sum_{i=1}^px_i\,|\, s=\sum_{i=1}^px_ia_i \textrm{ and } x_i\in \N \textrm{ for all }i\}.$ The Delta set of $S$, denoted by $\Delta(S)$, is the union of all the sets $\Delta(s)$ with $s\in S.$ As proved in [S.T. Chapman, R. Hoyer, and N. Kaplan. Delta sets of numerical monoids are eventually periodic. Aequationes Math. 77 (2009), no. 3, 273--279], there exists a bound $N$ such that $\Delta(S)$ is the union of the sets $\Delta(s)$ with $s\in S$ and $s<N$. In this work, by using geometrical tools, we obtain a sharpened bound and we give an algorithm to compute $\Delta(S)$ from the factorizations of only $a_1$ elements