Maria Vaz Pinto

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.

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.

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 combinatorial 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 vertex disjoint even cycles. In this case, a formula 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.

ON THE VANISHING IDEAL OF AN ALGEBRAIC TORIC SET AND ITS PARAMETRIZED 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).

Intertidal pools as alternative nursery habitats for coastal fishes

ABSTRACT Fish were sampled monthly in four tidal pools, for two years, on the west Portuguese coast. Species diversity of transient fish was higher than that found in previous studies, in other parts of the world. The transient fish population comprised six species: the white seabream, Diplodus sargus, sand smelt, Atherina spp., the thinlip grey mullet, Liza ramada, the Baillons wrasse, Symphodus bailloni, the zebra seabream, Diplodus cervinus and the European pilchard, Sardina pilchardus. Abundance varied seasonally, yearly, and among pools, with peak numbers in spring and summer. The most abundant species in all pools, both as larvae and juveniles, was D. sargus. Diplodus sargus and Atherina spp. were present in most pools, from spring to autumn, while rare species were present mostly in the spring-summer period. Smaller mean sizes of larvae and juveniles were observed at the beginning of spring of 2011 (March–April) and at the end of spring/beginning of summer of 2012 (May–June). Mean size of larvae and juveniles often showed a continuous increase from spring to autumn in both years. The highest density peaks were due to the high number of post-larvae entering the pools in spring. In most pools, the overall condition (Fultons K) of D. sargus increased throughout the year, in both years. The species richness, the high densities of early stages, and their continuous growth observed in tidal pools strongly emphasize the importance of these environments for larvae and juveniles of several transient marine fishes.

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.

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.

Commutative algebra : geometric, homological, combinatorial and computational aspects

A Theorem of Eakin and Sathaye and Greens Hyperplane Restriction Theorem. Liaison of Varieties of Small Dimension and Deficiency Modules. Regularity Jumps for Powers of Ideals. Integral Closure of Ideals and Annihilators of Homology. Poincare Series of Surface Singularities. Multi-Graded Hilbert Coefficients. Torsion Freeness and Normality of Blowup Rings of Monomial Ideals. Monomial Ideals via Square-Free Monomial Ideals. When Does the Equality I2=QI Hold True in Buchsbaum Rings? Integral Closures of Ideals in Completions of Regular Local Domains. The Components of a Variety of Matrices with Square Zero and Submaximal Rank. The Support of Top Graded Local Cohomology Modules. An Approach to the Uniform Artin-Rees Theorems from the Notion of Relation Type. Castelnuovo-Mumford Regularity and Finiteness of Hilbert Functions. Differential Idealizers and Algebraic Free Divisors. Gorenstein Rings Call the Tune. On Free Integral Extensions Generated by One Element. Degeneration of G-Dimension of Modules.