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Marcel Morales

Publications and source records attributed to Marcel Morales.

At least 19 recordsLinked to original sources
arXiv2026

Frobenius Number Of Almost Symmetric Numerical Generalized Almost Arithmetic Semigroups

Let a, k, h, c be positive integers and d a non zero integer. Recall that a numerical generalized almost arithmetic semigroup S is a semigroup minimally generated by relatively prime positive integers a, ha + d, ha + 2d, . . . , ha + kd, c, that is its embedding dimension is k + 2. In a previous work, the authors described the Ap{\'e}ry set and a Gr{\"o}bner basis of the ideal defining S under one technical assumption, the complete version will be published in a forthcoming paper. In this paper we continue with this assumption and we describe the Pseudo Frobenius set. As a consequence we give a complete description of S when it is symmetric or almost symmetric as well as generalize and extend the previous results of Ignacio Garc{\'i}a-Marco, J. L. Ram{\'i}rez Alfons{\'i}n and O. J. R{{\o}}dseth; we also find a quadratic formula for its Frobenius number that generalizes some results of J.C. Rosales, and P.A. Garc{\'i}a-S{\'a}nchez. Moreover, for given numbers a, d, k, h, c, a simple algorithm allows us to determine if S is almost symmetric or not and furthermore to find its type and Frobenius number.

math.AC↗
arXiv2024

Factor-critical graphs and dstab, astab for an edge ideal

Let $G$ be a simple, connected non bipartite graph and let $I_G$ be the edge idealof $G$. In our previous work we showed that L. Lov\'asz's theorem on ear decompositions offactor-critical graphs and the canonical decomposition of a graph given by Edmonds and Gallai are basic tools for the irreducible decomposition of $I^{k}_G$. In this paper we use some tools from graph theory, mainly Withney's theorem on ear decompositions of 2-edge connected graphs in order to introduce a new method to make a graph factor-critical. We can describe the set $\cup_ {k=1}^{\infty}{\rm Ass} (I^{k}_G)$ in terms of some subsets of $G$. We give explicit formulas for the numbers astab$(I_G)$ and dstab$(I_G)$, which are, respectively, the smallest number $k$ such that ${\rm Ass} (I^{k}_G)= {\rm Ass} (I^{k+i}_G)$ for all $i\geq 0$ and the smallest number $k$ such that the maximal ideal belongs to ${\rm Ass}(I^{k}_G)$. We also give very simple upper bounds for astab$(I_G)$ and dstab$(I_G)$.

math.AC↗
arXiv2017

Noether resolutions in dimension $2$

Let $R:= K[x_1,\ldots,x_{n}]$ be a polynomial ring over an infinite field $K$, and let $I \subset R$ be a homogeneous ideal with respect to a weight vector $ω= (ω_1,\ldots,ω_n) \in (\mathbb{Z}^+)^n$ such that $\dim(R/I) = d$. In this paper we study the minimal graded free resolution of $R/I$ as $A$-module, that we call the Noether resolution of $R/I$, whenever $A :=K[x_{n-d+1},\ldots,x_n]$ is a Noether normalization of $R/I$. When $d=2$ and $I$ is saturated, we give an algorithm for obtaining this resolution that involves the computation of a minimal Gröbner basis of $I$ with respect to the weighted degree reverse lexicographic order. In the particular case when $R/I$ is a $2$-dimensional semigroup ring, we also describe the multigraded version of this resolution in terms of the underlying semigroup. Whenever we have the Noether resolution of $R/I$ or its multigraded version, we obtain formulas for the corresponding Hilbert series of $R/I$, and when $I$ is homogeneous, we obtain a formula for the Castelnuovo-Mumford regularity of $R/I$. Moreover, in the more general setting that $R/I$ is a simplicial semigroup ring of any dimension, we provide its Macaulayfication. As an application of the results for $2$-dimensional semigroup rings, we provide a new upper bound for the Castelnuovo-Mumford regularity of the coordinate ring of a projective monomial curve. Finally, we describe the multigraded Noether resolution and the Macaulayfication of either the coordinate ring of a projective monomial curve $\mathcal{C} \subseteq \mathbb{P}_K^{n}$ associated to an arithmetic sequence or the coordinate ring of any canonical projection $π_{r}(\mathcal{C})$ of $\mathcal{C}$ to $\mathbb{P}_K^{n-1}$.

math.AC↗
arXiv2015

Gr{ö}bner basis. a "pseudo-polynomial" algorithm for computing the Frobenius number

Let consider $n$ natural numbers $a\_1 ,\ldots , a\_{n} $. Let $S$ be the numerical semigroup generated by $a\_1 ,\ldots , a\_{n} $. Set $A=K[t^{a\_1}, \ldots , t^{a\_n}]=K[{x\_1}, \ldots , {x\_n}]/I$. The aim of this paper is: \begin{enumerate}\item Give an effective pseudo-polynomial algorithm on $a\_1$, which computes The Ap{é}ry set and the Frobenius number of $S$. As a consequence it also solves in pseudo-polynomial time the integer knapsack problem : given a natural integer b, b belongs to $S$?\item The \gbb of $I$ for the reverse lexicographic order to $x\_n,\ldots ,x\_1$, without using Buchberger's algorithm. \item $\ini{I} $ for the reverse lexicographic order to $x\_n,\ldots ,x\_1$.\item $A$ as a $K[t^{ a\_1 }]$-module. \end{enumerate} We dont know the complexity of our algorithm. We need to solve the "multiplicative" integer knapsack problem: Find all positive integer solutions $({k\_1}, \ldots , {k\_n})$ of the inequality $\prod\_{i=2}^n (k\_i+1)\leq a\_1+1$. This algorithm is easily implemented. The implementation of this algorithm "frobenius-number-mm", for $n=17 $, can be downloaded in \hfill\breakhttps://www-fourier.ujf-grenoble.fr/~morales/frobenius-number-mm

math.AC↗
arXiv2014

A study of the length function of generalized fractions of modules

Let $(R, \frak m)$ be a Noetherian local ring and $M$ a finitely generated $R$-module of dimension $d$. Let $\underline{x} = x_1, ..., x_d$ be a system of parameters of $M$ and $\underline{n} = (n_1, ..., n_d)$ a $d$-tuple of positive integers. In this paper we study the length of generalized fractions $M (1/(x_1, ..., x_d, 1))$ which was introduced by Sharp and Hamieh in \cite{ShH85}. First, we study the growth of the function $J_{\underline{x}, M}(\underline{n}) = \ell(M (1/(x_1^{n_1}, ..., x_d^{n_d}, 1))) - n_1...n_d e(\underline{x};M)$. Then we give an explicit calculation for the function $J_{\underline{x}, M}(\underline{n})$ in the case where $M$ admits a Macaulayfication. Most previous results on this topic are now easy to understand and to improve.

math.AC↗
arXiv2014

Hilbert series of Segre transform, and Castelnuovo-Mumford regularity

In a recent preprint, Ilse Fischer and Martina Kubitzke, proved the bilinearity of the Segre transform under some restricted hypothesis, motivated by their results we show in this paper the bilinearity of the Segre transform in general. We apply these results to compute the postulation number of a series. Our second application is motivated by the paper of David A. Cox, and Evgeny Materov (2009), where is computed the Castelnuovo-Mumford regularity of the Segre Veronese embedding, we can extend partially their result and compute the Castelnuovo-Mumford regularity of the Segre product of Cohen-Macaulay modules.

math.AC↗
arXiv2014

Segre embeddings, Hilbert series and Newcomb's problem

Monomial ideals and toric rings are closely related. By consider a Grobner basis we can always associated to any ideal $I$ in a polynomial ring a monomial ideal ${\rm in}_\prec I$, in some special situations the monomial ideal ${\rm in}_\prec I$ is square free. On the other hand given any monomial ideal $I$ of a polynomial ring $S$, we can define the toric $K[I]\subset S$. In this paper we will study toric rings defined by Segre embeddings, we will prove that their $h-$ vectors coincides with the so called Simon Newcomb number's in probabilities and combinatorics. We solve the original question of Simon Newcomb by given a formula for the Simon Newcomb's numbers involving only positive integer numbers.

math.AC↗
arXiv2012

The ${\rm N}_{2,p}$-property of binomial extensions of simplicial complexes

M. Morales introduced a family of binomial ideals that are binomial extensions of square free monomial ideals. Let $I\subset \si$ be a square free monomial ideal and $J\subset\sis$ a sum of scroll ideals with some extra conditions, we define the binomial extension of $I$ as $\B=I+J\subset \sis$. We set $p_2(\B)$ the minimal $i\in\N$ such that there exists $j>2$ such that $β_{i,i+j}(\B)\neq 0$. In the case where J=0, Fröberg characterized combinatorally the case $p_2(I)=\infty$; later Eisenbud et al. solved the case $p_2(I)<\infty$. We obtain a similar result as Fröberg for the binomial extensions and we find lower and upper bounds of $p_2(\B)$ for some families of binomial extensions in combinatorial terms as Eisenbud et al. With some additional hypothesis we can compute $p_2(\B)$.

math.AC↗
arXiv2012

Sums of toric ideals

Given two toric ideals $I_1,I_2\subset\si$, it is not always true that $I_1+I_2$ is a toric ideal. Given $I_1,...,I_k\subset\si$ a familly of toric ideals we give necessary conditions in order to have that $I_1+...+I_k$ is a toric ideal.

math.AC↗
arXiv2012

Regularity of edge ideal of a graph

In this paper, we introduce some reduction processes on graphs which preserve the regularity of related edge ideals. As a consequence, an alternative proof for the theorem of R. Fröberg on linearity of resolution of edge ideal of graphs is given.

math.AC↗
arXiv2012

Regularity and Free Resolution of Ideals which are Minimal to $d$-linearity

Toward a partial classification of monomial ideals with $d$-linear resolution, in this paper, some classes of $d$-uniform clutters which do not have linear resolution, but every proper subclutter of them has a $d$-linear resolution, are introduced and the regularity and Betti numbers of circuit ideals of such clutters are computed. Also, it is proved that for given two $d$-uniform clutters $\mathcal{C}_1, \mathcal{C}_2$, the Castelnuovo-Mumford regularity of the ideal $I(\bar{\mathcal{C}_1 \cup \mathcal{C}_2})$ is equal to the maximum of regularities of $I(\bar{\C}_1)$ and $I(\bar{\C}_2)$, whenever $V(\mathcal{C}_1) \cap V(\mathcal{C}_2)$ is a clique or ${\rm SC}(\mathcal{C}_1) \cap {\rm SC}(\mathcal{C}_2)=\emptyset$. As applications, alternative proofs are given for Fröberg's Theorem on linearity of edge ideal of graphs with chordal complement as well as for linearity of generalized chordal hypergraphs defined by Emtander. Finally, we find minimal free resolutions of the circuit ideal of a triangulation of a pseudo-manifold and a homology manifold explicitly.

math.AC↗
arXiv2012

Monomial ideals with 3-linear resolutions

In this paper, we study Cstelnuovo-Mumford regularity of square-free monomial ideals generated in degree 3. We define some operations on the clutters associated to such ideals and prove that the regularity is conserved under these operations. We apply the operations to introduce some classes of ideals with linear resolutions and also show that any clutter corresponding to a triangulation of the sphere does not have linear resolution while any proper sub-clutter of it has a linear resolution.

math.AC↗
arXiv2010

Binomial generation of the radical of a lattice ideal

Let $I_{L, ρ}$ be a lattice ideal. We provide a necessary and sufficient criterion under which a set of binomials in $I_{L, ρ}$ generate the radical of $I_{L, ρ}$ up to radical. We apply our results to the problem of determining the minimal number of generators of $I_{L, ρ}$ or of the $rad(I_{L, ρ})$ up to radical.

math.AC↗
arXiv2009

Binomial extensions of Simplicial ideals and reduction number

In this article, we define a class of binomial ideals associated to a simplicial complex. This class of ideals appears in the presentation of fiber cones of codimension 2 lattice ideals \cite{hm}, and in the work of Barile and Morales \cite{bm2}, \cite{bm3}, \cite{bm4}. We compute the reduction number of Binomial extensions of Simplicial ideals. This extends all the previous results in this area.

math.AC↗
arXiv2008

$p-$Ferrer diagram, $p-$linear ideals and arithmetical rank

In this paper we introduce $p-$Ferrer diagram, note that $1-$ Ferrer diagram are the usual Ferrer diagrams or Ferrer board, and corresponds to planar partitions. To any $p-$Ferrer diagram we associate a $p-$Ferrer ideal. We prove that $p-$Ferrer ideal have Castelnuovo mumford regularity $p+1$. We also study Betti numbers, minimal resolutions of $p-$Ferrer ideals. Every $p-$Ferrer ideal is $p-$joined ideals in a sense defined in a fortcoming paper \cite{m2}, which extends the notion of linearly joined ideals introduced and developped in the papers \cite{bm2}, \cite{bm4},\cite{eghp} and \cite{m1}. We can observe the connection between the results on this paper about the Poincaré series of a $p-$Ferrer diagram $Φ$and the rook problem, which consist to put $k$ rooks in a non attacking position on the $p-$Ferrer diagram $Φ$.

math.AC↗
arXiv2008

Equations of 2-linear ideals and arithmetical rank

In this paper we consider reduced homogeneous ideals $\Jcal\subset S$ of a polynomial ring $S$, having a 2-linear resolution. 1. We study systems of generators of $\Jcal\subset S$. 2. We compute the arithmetical rank for a large class of projective curves having a 2-linear resolution. 3. We show that the fiber cone $\proj \Fcal(I_{\Lcal})$ of a lattice ideal $I_{\Lcal}$ of codimension two is a set theoretical complete intersection.

math.AC↗
arXiv2007

Simplicial ideals, 2-linear ideals and arithmetical rank

In the first part of this paper we study scrollers and linearly joined varieties. A particular class of varieties, of important interest in classical Geometry are Cohen--Macaulay varieties of minimal degree. They appear naturally studying the fiber cone of of a codimension two toric ideals. Let $I\subset S$ be an ideal defining a linearly joined arrangement of varieties: - We compute the depth, and the cohomological dimension. is the connectedness dimension. - We characterize sets of generators of $I$, and give an effective algorithm to find equations, as an application we compute arithmetical rank. in the case if $I$ defines a union of linear spaces, (ara =projective dimension), in particular this applies to any square free monomial ideal having a $2-$ linear resolution. - In the case where $V$ is a union of linear spaces, the ideal $I$, can be characterized by a tableau, which is an extension of a Ferrer (or Young) tableau. - We introduce a new class of ideals called simplicial ideals, ideals defining linearly-joined varieties are a particular case of simplicial ideals.

math.AC↗