Ignacio Ojeda
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View article: The multiples of a numerical semigroup
The multiples of a numerical semigroup Open
Given two numerical semigroups $S$ and $T$ we say that $T$ is a multiple of $S$ if there exists an integer $d \in \mathbb{N} \setminus \{0\}$ such that $S = \{x \in \mathbb{N} \mid d x \in T\}$. In this paper we study the family of multipl…
View article: On the depth of simplicial affine semigroup rings
On the depth of simplicial affine semigroup rings Open
We recall and delve into the different characterizations of the depth of an affine semigroup ring, providing an original characterization of depth two in three and four dimensional cases which are closely related to the existence of a maxi…
View article: Minimal free resolution of generalized repunit algebras
Minimal free resolution of generalized repunit algebras Open
Let $\Bbbk$ be an arbitrary field and let $b > 1, n > 1$ and $a$ be three positive integers. In this paper we explicitly describe a minimal $S-$graded free resolution of the semigroup algebra $\Bbbk[S]$ when $S$ is a generalized repunit nu…
View article: Arithmetic varieties of numerical semigroups
Arithmetic varieties of numerical semigroups Open
In this paper we present the notion of arithmetic variety for numerical semigroups. We study various aspects related to these varieties such as the smallest arithmetic that contains a set of numerical semigroups and we exhibit the root thr…
View article: Stability of singular limit cycles for Abel equations revisited
Stability of singular limit cycles for Abel equations revisited Open
A criterion is obtained for the semi-stability of the isolated singular positive closed solutions, i.e., singular positive limit cycles, of the Abel equation x′=A(t)x3+B(t)x2, where A,B are smooth functions with two zeros in the interval […
View article: On the depth of simplicial affine semigroup rings
On the depth of simplicial affine semigroup rings Open
We recall and delve into the different characterizations of the depth of an affine semigroup ring, providing an original characterization of depth two in three and four dimensional cases which are closely related to the existence of a maxi…
View article: Hilbert number for a family of piecewise nonautonomous equations
Hilbert number for a family of piecewise nonautonomous equations Open
For family $x'=(a_0+a_1\cos t+a_2 \sin t)|x|+b_0+b_1 \cos t+b_2 \sin t$, we solve three basic problems related with its dynamics. First, we characterize when it has a center (Poincaré center focus problem). Second, we show that each equati…
View article: Universally free numerical semigroups
Universally free numerical semigroups Open
A numerical semigroup is said to be universally free if it is free for any possible arrangement of its minimal generating set. In this work, we establish that toric ideals associated with universally free numerical semigroups can be genera…
View article: Rational Limit Cycles of Abel Differential Equations
Rational Limit Cycles of Abel Differential Equations Open
We study the number of rational limit cycles of the Abel equation $x'=A(t)x^3+B(t)x^2$, where $A(t)$ and $B(t)$ are real trigonometric polynomials. We show that this number is at most the degree of $A(t)$ plus one.
View article: Minimal Systems of Binomial Generators for the Ideals of Certain Monomial Curves
Minimal Systems of Binomial Generators for the Ideals of Certain Monomial Curves Open
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 determina…
View article: The Frobenius problem for generalized repunit numerical semigroups
The Frobenius problem for generalized repunit numerical semigroups Open
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 …
View article: Minimal Systems of Binomial Generators for the Ideals of Certain Monomial Curves
Minimal Systems of Binomial Generators for the Ideals of Certain Monomial Curves Open
Let $a, b$ and $n > 1$ be three positive integers such that $a$ and $\sum_{j=0}^{n-1} b^j$ are relatively prime. In this paper, we prove that the toric ideal $I$ associated to the submonoid of $\mathbb{N}$ generated by $\{\sum_{j=0}^{n-…
View article: Minimal binomial systems of generators for the ideals of certain monomial curves
Minimal binomial systems of generators for the ideals of certain monomial curves Open
Let $a, b$ and $n > 1$ be three positive integers such that $a$ and $\sum_{j=0}^{n-1} b^j$ are relatively prime. In this paper, we prove that the toric ideal $I$ associated to the submonoid of $\mathbb{N}$ generated by $\{\sum_{j=0}^{n-1} …
View article: The set of numerical semigroups of a given multiplicity and Frobenius number
The set of numerical semigroups of a given multiplicity and Frobenius number Open
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 se…
View article: A Note on Decomposable and Reducible Integer Matrices
A Note on Decomposable and Reducible Integer Matrices Open
We propose necessary and sufficient conditions for an integer matrix to be decomposable in terms of its Hermite normal form. Specifically, to each integer matrix, we associate a symmetric integer matrix whose reducibility can be efficientl…
View article: Renovation of the Traditional Concert Practice: Transforming the Conventional Concert into a New Artistic Experience
Renovation of the Traditional Concert Practice: Transforming the Conventional Concert into a New Artistic Experience Open
In this paper, the author tries to address the huge divide between the classical music world and its institutions from a potential audience who is culturally interested but also engaged with other artforms which place their focus on the pr…
View article: On pseudo-Frobenius elements of submonoids of $$\mathbb {N}^d$$
On pseudo-Frobenius elements of submonoids of $$\mathbb {N}^d$$ Open
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 po…
View article: The set of numerical semigroups of a given multiplicity and Frobenius\n number
The set of numerical semigroups of a given multiplicity and Frobenius\n number Open
We study the structure of the family of numerical semigroups with fixed\nmultiplicity and Frobenius number. We give an algorithmic method to compute all\nthe semigroups in this family. As an application we compute the set of all\nnumerical…
View article: On pseudo-Frobenius elements of submonoids of $\mathbb{N}^d$
On pseudo-Frobenius elements of submonoids of $\mathbb{N}^d$ Open
In this paper we study those submonoids of $\mathbb{N}^d$ which a non-trivial 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 pos…
View article: Almost symmetric numerical semigroups with high type
Almost symmetric numerical semigroups with high type Open
We establish a one-to-one correspondence between numerical semigroups of\ngenus $g$ and almost symmetric numerical semigroups with Frobenius number $F$\nand type $F-2g$, provided that $F$ is greater than $4g-1$.\n
View article: Almost symmetric numerical semigroups with given Frobenius number and type
Almost symmetric numerical semigroups with given Frobenius number and type Open
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 algorith…
View article: Uniqueness of limit cycles for quadratic vector fields
Uniqueness of limit cycles for quadratic vector fields Open
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 $x'= a_1 x-y-a_3x^2+(2 a_2+a_5)xy + a_6 y^2$, $y'= x+a_1 y + a_2x^2+(…
View article: Almost symmetric numerical semigroups with given Frobenius number and type
Almost symmetric numerical semigroups with given Frobenius number and type Open
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 algorith…
View article: THE SHORT RESOLUTION OF A SEMIGROUP ALGEBRA
THE SHORT RESOLUTION OF A SEMIGROUP ALGEBRA Open
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 provide…
View article: Critical binomial ideals of Norhtcott type
Critical binomial ideals of Norhtcott type Open
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 $x_i^{c_i} - x_1^{u_{i1}} \cdots x_n^{u_{1n}} \in k[x_1, \ldots, x_n]$ with $u_{ii…