arXiv subjects

Maria Vaz Pinto

Publications and source records attributed to Maria Vaz Pinto.

17 recordsLinked to original sources

arXiv2026

On the regularity index of the minimum distance function in projective nested Cartesian codes

Let $X$ be a projective nested product of fields and let $\delta_X(d)$ be the minimum distance in degree $d\geq 1$ of the projective nested Cartesian code $C_X(d)$. The regularity index ${\rm reg}(\delta_X)$ of the minimum distance function $\delta_X$ is the minimum integer $d_0\geq 0$ such that $\delta_X(d)=1$ for $d\geq d_0$. We give a formula for ${\rm reg}(\delta_X)$ by determining an indicator function of least degree for each point of $X$ and using the fact that ${\rm reg}(\delta_X)$ is the ${\rm v}$-number of the vanishing ideal $I_X$ of $X$. Then we give an arithmetical criterion that characterizes when $X$ is Cayley--Bacharach.

math.AC↗

arXiv2025

Graph rings and ideals: Wolmer Vasconcelos' contributions

This is a survey article featuring some of Wolmer Vasconcelos' contributions to commutative algebra, and explaining how Vasconcelos' work and insights have contributed to the development of commutative algebra and its interaction with other areas to the present. We discuss the Vasconcelos' function and the Vasconcelos' number (v-number for short) of graded ideals and their relation to coding theory, and the interplay of Simis and normal monomial ideals with combinatorial optimization problems, blowup algebras, and resurgence theory. The regularity of subrings of normal k-uniform monomial ideals is shown to be a monotone function, and we give a normality criterion for edge ideals of graphs using Ehrhart rings.

math.AC↗

arXiv2024

Symbolic powers: Simis and weighted monomial ideals

The aim of this work is to compare symbolic and ordinary powers of monomial ideals using commutative algebra and combinatorics. Monomial ideals whose symbolic and ordinary powers coincide are called Simis ideals. Weighted monomial ideals are defined by assigning linear weights to monomials. We examine Simis and normally torsion-free ideals, relate some of the properties of monomial ideals and weighted monomial ideals, and present a structure theorem for edge ideals of $d$-uniform clutters whose ideal of covers is Simis in degree $d$. One of our main results is a combinatorial classification of when the dual of the edge ideal of a weighted oriented graph is Simis in degree $2$.

math.AC↗

arXiv2021

Parameterized codes over graphs

In this article we review known results on parameterized linear codes over graphs, introduced by Rentería, Simis and Villarreal in 2011. Very little is known about their basic parameters and invariants. We review in detail the parameters dimension, regularity and minimum distance. As regards the parameter dimension, we explore the connection to Eulerian ideals in the ternary case and we give new combinatorial formulas.

math.AC↗

arXiv2020

Evaluation codes and their basic parameters

The aim of this work is to give degree formulas for the generalized Hamming weights of evaluation codes and to show lower bounds for these weights. In particular, we give degree formulas for the generalized Hamming weights of Reed--Muller-type codes, and we determine the minimum distance of toric codes over hypersimplices, and the 1st and 2nd generalized Hamming weights of squarefree evaluation codes.

math.AC↗

arXiv2019

Generalized minimum distance functions and algebraic invariants of Geramita ideals

Motivated by notions from coding theory, we study the generalized minimum distance (GMD) function $δ_I(d,r)$ of a graded ideal $I$ in a polynomial ring over an arbitrary field using commutative algebraic methods. It is shown that $δ_I$ is non-decreasing as a function of $r$ and non-increasing as a function of $d$. For vanishing ideals over finite fields, we show that $δ_I$ is strictly decreasing as a function of $d$ until it stabilizes. We also study algebraic invariants of Geramita ideals. Those ideals are graded, unmixed, $1$-dimensional and their associated primes are generated by linear forms. We also examine GMD functions of complete intersections and show some special cases of two conjectures of Tohăneanu--Van Tuyl and Eisenbud-Green-Harris.

math.AC↗

arXiv2019

Joins, Ears and Castelnuovo-Mumford regularity

We introduce a new class of polynomial ideals associated to a simple graph, $G$. Let $K[E_G]$ be the polynomial ring on the edges of $G$ and $K[V_G]$ the polynomial ring on the vertices of $G$. We associate to $G$ an ideal, $I(X_G)$, defined as the preimage of $(x_i^2-x_j^2 : i,j\in V_G)\subseteq K[V_G]$ by the map $K[E_G]\to K[V_G]$ which sends a variable, $t_e$, associated to an edge $e=\{i,j\}$, to the product $x_ix_j$ of the variables associated to its vertices. We show that $K[E_G]/I(X_G)$ is a one-dimensional, Cohen-Macaulay, graded ring, that $I(X_G)$ is a binomial ideal and that, with respect to a fixed monomial order, its initial ideal has a generating set independent of the field $K$. We focus on the Castelnuovo-Mumford regularity of $I(X_G)$ providing the following sharp upper and lower bounds: $$ μ(G) \leq \operatorname{reg} I(X_G) \leq |V_G|-b_0(G)+1, $$ where $μ(G)$ is the maximum vertex join number of the graph and $b_0(G)$ is the number of its connected components. We show that the lower bound is attained for a bipartite graph and use this to derive a new combinatorial result on the number of even length ears of nested ear decomposition.

math.AC↗

arXiv2016

Regularity of the vanishing ideal over a parallel composition of paths

Let G be a graph obtained by taking r>=2 paths and identifying all first vertices and identifying all the last vertices. We compute the Castelnuovo--Mumford regularity of the quotient S/I(X), where S is the polynomial ring on the edges of G and I(X) is the vanishing ideal of the projective toric subset parameterized by G. The case we consider is the first case where the regularity was unknown, following earlier computations (by several authors) of the regularity when G is a tree, cycle, complete graph or complete bipartite graph, but specially in light of the reduction of the computation of the regularity in the bipartite case to the computation of the regularity of the blocks of G. We also prove new inequalities relating the Castelnuovo--Mumford regularity of S/I(X) with the combinatorial structure of G, for a general graph.

math.AC↗

arXiv2016

Direct products in projective Segre codes

Let K=Fq be a finite field. We introduce a family of projective Reed-Muller-type codes called projective Segre codes. Using commutative algebra and linear algebra methods, we study their basic parameters and show that they are direct products of projective Reed-Muller-type codes. As a consequence we recover some results on projective Reed-Muller-type codes over the Segre variety and over projective tori.

math.AC↗

arXiv2013

Regularity and algebraic properties of certain lattice ideals

We study the regularity and the algebraic properties of certain lattice ideals. We establish a map I --> I\~ between the family of graded lattice ideals in an N-graded polynomial ring over a field K and the family of graded lattice ideals in a polynomial ring with the standard grading. This map is shown to preserve the complete intersection property and the regularity of I but not the degree. We relate the Hilbert series and the generators of I and I\~. If dim(I)=1, we relate the degrees of I and I\~. It is shown that the regularity of certain lattice ideals is additive in a certain sense. Then, we give some applications. For finite fields, we give a formula for the regularity of the vanishing ideal of a degenerate torus in terms of the Frobenius number of a semigroup. We construct vanishing ideals, over finite fields, with prescribed regularity and degree of a certain type. Let X be a subset of a projective space over a field K. It is shown that the vanishing ideal of X is a lattice ideal of dimension 1 if and only if X is a finite subgroup of a projective torus. For finite fields, it is shown that X is a subgroup of a projective torus if and only if X is parameterized by monomials. We express the regularity of the vanishing ideal over a bipartie graph in terms of the regularities of the vanishing ideals of the blocks of the graph.

math.AC↗

arXiv2013

Vanishing ideals over complete multipartite graphs

We study the vanishing ideal of the parametrized algebraic toric associated to the complete multipartite graph $\G=\mathcal{K}_{α_1,...,α_r}$ over a finite field of order $q$. We give an explicit family of binomial generators for this lattice ideal, consisting of the generators of the ideal of the torus, (referred to as type I generators), a set of quadratic binomials corresponding to the cycles of length 4 in $\G$ and which generate the \emph{toric algebra of $\G$} (type II generators) and a set of binomials of degree $q-1$ obtained combinatorially from $\G$ (type III generators). Using this explicit family of generators of the ideal, we show that its Castelnuovo--Mumford regularity is equal to $\max\set{α_1(q-2),...,α_r(q-2), \lceil (n-1)(q-2)/2\rceil}$, where $n=α_1+... + α_r$.

math.AC↗

arXiv2013

The Degree and regularity of vanishing ideals of algebraic toric sets over finite fields

Let X* be a subset of an affine space A^s, over a finite field K, which is parameterized by the edges of a clutter. Let X and Y be the images of X* under the maps x --> [x] and x --> [(x,1)] respectively, where [x] and [(x,1)] are points in the projective spaces P^{s-1} and P^s respectively. For certain clutters and for connected graphs, we were able to relate the algebraic invariants and properties of the vanishing ideals I(X) and I(Y). In a number of interesting cases, we compute its degree and regularity. For Hamiltonian bipartite graphs, we show the Eisenbud-Goto regularity conjecture. We give optimal bounds for the regularity when the graph is bipartite. It is shown that X* is an affine torus if and only if I(Y) is a complete intersection. We present some applications to coding theory and show some bounds for the minimum distance of parameterized linear codes for connected bipartite graphs.

math.AC↗

arXiv2012

Vanishing ideals over graphs and even cycles

Let X be an algebraic toric set in a projective space over a finite field. We study the vanishing ideal, I(X), of X and show some useful degree bounds for a minimal set of generators of I(X). We give an explicit description of a set of generators of I(X), when X is the algebraic toric set associated to an even cycle or to a connected bipartite graph with pairwise disjoint even cycles. In this case, a fomula for the regularity of I(X) is given. We show an upper bound for this invariant, when X is associated to a (not necessarily connected) bipartite graph. The upper bound is sharp if the graph is connected. We are able to show a formula for the length of the parameterized linear code associated with any graph, in terms of the number of bipartite and non-bipartite components.

math.AC↗

arXiv2011

Parameterized affine codes

Let K be a finite field and let X* be an affine algebraic toric set parameterized by monomials. We give an algebraic method, using Groebner bases, to compute the length and the dimension of C_X*(d), the parameterized affine code of degree d on the set X*. If Y is the projective closure of X*, it is shown that C_X^*(d) has the same basic parameters that C_Y(d), the parameterized projective code on the set Y. If X* is an affine torus, we compute the basic parameters of C_X*(d). We show how to compute the vanishing ideals of X* and Y.

math.AC↗

arXiv2011

On the vanishing ideal of an algebraic toric set and its parameterized linear codes

Let K be a finite field and let X be a subset of a projective space, over the field K, which is parameterized by monomials arising from the edges of a clutter. We show some estimates for the degree-complexity, with respect to the revlex order, of the vanishing ideal I(X) of X. If the clutter is uniform, we classify the complete intersection property of I(X) using linear algebra. We show an upper bound for the minimum distance of certain parameterized linear codes along with certain estimates for the algebraic invariants of I(X).

math.AC↗

arXiv2011

The minimum distance of parameterized codes on projective tori

Let X be a subset of a projective space, over a finite field K, which is parameterized by the monomials arising from the edges of a clutter. Let I(X) be the vanishing ideal of X. It is shown that I(X) is a complete intersection if and only if X is a projective torus. In this case we determine the minimum distance of any parameterized linear code arising from X.

math.AC↗

arXiv2002

Sally modules and associated graded rings

We study the depth properties of the associated graded ring of an m-primary ideal I in terms of numerical data attached to the ideal I. We also find bounds on the Hilbert coefficients of I by means of the Sally module S_J(I) of I with respect to a minimal reduction J of I.

math.AC↗