Philippe Giménez
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View article: Projective Cohen-Macaulay monomial curves and their affine charts
Projective Cohen-Macaulay monomial curves and their affine charts Open
In this paper, we explore when the Betti numbers of the coordinate rings of a projective monomial curve and one of its affine charts are identical. Given an infinite field k and a sequence of relatively prime integers $$a_0 = 0< a_1< \cdot…
View article: Sifted degrees of the equations of the Rees module and their connection with the Artin-Rees numbers
Sifted degrees of the equations of the Rees module and their connection with the Artin-Rees numbers Open
Let $A$ be a noetherian ring, $I$ an ideal of $A$ and $N\subset M$ finitely generated $A$-modules. The relation type of $I$ with respect to $M$, denoted by ${\bf rt}\,(I;M)$, is the maximal degree in a minimal generating set of relations o…
View article: Proyective Cohen-Macaulay monomial curves and their affine charts
Proyective Cohen-Macaulay monomial curves and their affine charts Open
In this paper, we explore when the Betti numbers of the coordinate rings of a projective monomial curve and one of its affine charts are identical. Given an infinite field $k$ and a sequence of relatively prime integers $a_0 = 0 < a_1 < \c…
View article: Gluing And Splitting of Homogeneous Toric Ideals
Gluing And Splitting of Homogeneous Toric Ideals Open
We show that any two homogeneous affine semigroups can be glued by embedding them suitably in a higher dimensional space. As a consequence, we show that the sum of their homogeneous toric ideals is again a homogeneous toric ideal, and that…
View article: Subfield subcodes of projective Reed-Muller codes
Subfield subcodes of projective Reed-Muller codes Open
Explicit bases for the subfield subcodes of projective Reed-Muller codes over the projective plane and their duals are obtained. In particular, we provide a formula for the dimension of these codes. For the general case over the projective…
View article: Entanglement-assisted quantum error-correcting codes from subfield subcodes of projective Reed–Solomon codes
Entanglement-assisted quantum error-correcting codes from subfield subcodes of projective Reed–Solomon codes Open
We study the subfield subcodes of projective Reed–Solomon codes and their duals: we provide bases for these codes and estimate their parameters. With this knowledge, we can construct symmetric and asymmetric entanglement-assisted quantum e…
View article: Subfield subcodes of projective Reed-Muller codes
Subfield subcodes of projective Reed-Muller codes Open
Explicit bases for the subfield subcodes of projective Reed-Muller codes over the projective plane and their duals are obtained. In particular, we provide a formula for the dimension of these codes. For the general case over the projective…
View article: Castelnuovo-Mumford regularity of projective monomial curves via sumsets
Castelnuovo-Mumford regularity of projective monomial curves via sumsets Open
Let $A=\{a_0,\ldots,a_{n-1}\}$ be a finite set of $n\geq 4$ non-negative relatively prime integers such that $0=a_0
View article: Entanglement-assisted quantum error-correcting codes from subfield subcodes of projective Reed-Solomon codes
Entanglement-assisted quantum error-correcting codes from subfield subcodes of projective Reed-Solomon codes Open
We study the subfield subcodes of projective Reed-Solomon codes and their duals: we provide bases for these codes and estimate their parameters. With this knowledge, we can construct symmetric and asymmetric entanglement-assisted quantum e…
View article: Saturation and vanishing ideals
Saturation and vanishing ideals Open
We consider an homogeneous ideal $I$ in the polynomial ring $S=K[x_1,\dots,$ $x_m]$ over a finite field $K=\mathbb{F}_q$ and the finite set of projective rational points $\mathbb{X}$ that it defines in the projective space $\mathbb{P}^{m-1…
View article: On gluing semigroups in $\mathbb{N}^n$ and the consequences
On gluing semigroups in $\mathbb{N}^n$ and the consequences Open
A semigroup $\langle C\rangle$ in $\mathbb{N}^n$ is a gluing of $\langle A\rangle$ and $\langle B\rangle$ if its finite set of generators $C$ splits into two parts, $C=k_1A\sqcup k_2B$ with $k_1,k_2\geq 1$, and the defining ideals of the c…
View article: Commutative Algebra: Geometric, Homological, Combinatorial and Computational Aspects
Commutative Algebra: Geometric, Homological, Combinatorial and Computational Aspects Open
A Theorem of Eakin and Sathaye and Green's 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. Poin…
View article: Symbolic powers of monomial ideals and Cohen-Macaulay vertex-weighted digraphs
Symbolic powers of monomial ideals and Cohen-Macaulay vertex-weighted digraphs Open
In this paper we study irreducible representations and symbolic Rees algebras of monomial ideals. Then we examine edge ideals associated to vertex-weighted oriented graphs. These are digraphs having no oriented cycles of length two with we…
View article: On complete monomial ideals
On complete monomial ideals Open
In dimension two, we study complete monomial ideals combinatorially, their\nRees algebras and develop effective means to find their defining equations.\n