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We make explicit the exponential bound on the degrees of the polynomials appearing in the Effective Quillen-Suslin Theorem, and apply it jointly with the Hilbert-Burch Theorem to show that the syzygy module of a sequence of $m$ polynomials in $n$ variables defining a complete intersection ideal of grade two is free, and that a basis of it can be computed with bounded degrees. In the known cases, these bounds improve previous results.

Treballs Finals de Grau de Matemàtiques, Facultat de Matemàtiques, Universitat de Barcelona, Any: 2022, Director: Teresa Crespo Vicente, [en] Abstract It's one of the first results a maths undergraduate hears about. The function $e^{-x^{2}}$ does not have an antiderivative, the value of its integral on any given interval where it can be calculated can only be approximated via numerical methods. But what does it mean for a real function to not have an antiderivative, a function that expresses its integral in simple terms? When do we even consider a real or complex function to be expressible in simple terms? These questions are the focus of this project. Using modern results in mathematics, mainly differential algebra, we aim to introduce a theoretical frame where these questions can be posed rigorously, one where we can prove the theorem that answers them: Liouville's theorem about antiderivatives.

Treballs Finals de Grau de Matemàtiques, Facultat de Matemàtiques, Universitat de Barcelona, Any: 2022, Director: Santiago Zarzuela, [en] In this work, we study some of the correspondences between commutative algebra and graph theory. We start with an introduction on edge and cover ideals that lead into introducing the algebra basic tools that lets us state Fröberg’s Theorem on linear resolutions of edge ideals and chordal graphs. The work ends with a proof of Fröberg’s Theorem given by Adam Van Tuyl, based on monomial splittings and some properties of chordal graphs.

Given any diagonal cyclic subgroup $\Lambda \subset GL(n+1,k)$ of order $d$, let $I_d\subset k[x_0,\ldots, x_n]$ be the ideal generated by all monomials $\{m_{1},\ldots, m_{r}\}$ of degree $d$ which are invariants of $\Lambda$. $I_d$ is a monomial Togliatti system, provided $r \leq \binom{d+n-1}{n-1}$, and in this case the projective toric variety $X_d$ parameterized by $(m_{1},\ldots, m_{r})$ is called a $GT$-variety with group $\Lambda$. We prove that all these $GT$-varieties are arithmetically Cohen-Macaulay and we give a combinatorial expression of their Hilbert functions. In the case $n=2$, we compute explicitly the Hilbert function, polynomial and series of $X_d$. We determine a minimal free resolution of its homogeneous ideal and we show that it is a binomial prime ideal generated by quadrics and cubics. We also provide the exact number of both types of generators. Finally, we pose the problem of determining whether a surface parameterized by a Togliatti system is aCM. We construct examples that are aCM and examples that are not., Comment: To appear in Annali di Matematica Pura ed Applicata. Minor correction in the Introduction

Treballs Finals de Grau de Matemàtiques, Facultat de Matemàtiques, Universitat de Barcelona, Any: 2022, Director: Carlos D'Andrea, [en] The main goal of this work is to study polynomial interpolation in several variables from an algebraic perspective. To do so, we treat linear differential operators as algebraic elements, and consider the solution space of a polynomial interpolation problem as the orthogonal space via a sesqui-linear map of an ideal of multivariate polynomials. Examples and a Mathematica code are also provided.

Treballs Finals de Grau de Matemàtiques, Facultat de Matemàtiques, Universitat de Barcelona, Any: 2022, Director: Rosa M. Miró-Roig, [en] The main purpose of this paper is to study the so-called sumsets problem. This problem is naturally seen from the point of view of Additive Combinatorics, yet we approach it using Algebraic Geometry. This work is divided into three chapters. The first chapter is devoted to Commutative Algebra. We first define basic concepts, such as graded modules or exact sequences, which will be present throughout the whole article, and then we introduce the concept of the Hilbert function of a graded module. The most important result of the chapter is the fact that this function, for sufficiently large integers, is a polynomial, which we prove by means of the Hilbert-Serre theorem and also Hilbert’s syzygy theorem. Knowing the coefficients of this polynomial is, in general, a very difficult problem. In the second chapter, we link the previous one with Algebraic Geometry. We define the Hilbert function of a projective variety and we calculate it in some simple cases. Next, we study three invariants of projective varieties and introduce the Veronese varieties, which are key in this work. The monomial projections of these varieties will be fundamental to solving the sumsets problem. Finally, in the last chapter, we show that the cardinality of the sumsets can be modeled by the Hilbert function of a suitable monomial projection of a Veronese variety, which proves that this cardinality asymptotically becomes a polynomial.

Treballs Finals de Grau de Matemàtiques, Facultat de Matemàtiques, Universitat de Barcelona, Any: 2022, Director: Santiago Zarzuela, [en] The Fundamental Theorem of Finitely Generated Abelian Groups is a very important result that allows us to describe explicitly the finitely generated abelian groups. This theorem can naturally be generalized to finitely generated modules over principal ideal domains. The difficulty arises when we try to extend it to other types of rings. In this work we prove a generalization of this, the decomposition theorem of finitely generated modules over Dedekind domains. We also explore other similar decomposition and we characterize the rings such that all their modules have a decomposition of simple and cyclic modules.

Treballs Finals de Grau de Matemàtiques, Facultat de Matemàtiques, Universitat de Barcelona, Any: 2022, Director: Artur Travesa i Grau, [en] Since the birth of the AES, ways have been sought to try to break it. It seems that the most common cryptoanalytic methods do not succeed. For this reason the XSL is invented, an algebraic attack against the AES. The goal of this thesis is to study the algebra of this attack.

In this paper, we introduce the notion of a complete hypertetrahedral arrangement $\mathcal{A}$ in $\mathbb{P}^{n}$. We address two basic problems. First, we describe the local freeness of $\mathcal{A}$ in terms of smaller complete hypertetrahedral arrangements and graph theory properties, specializing the Musta\c{t}\u{a}-Schenck criterion. As an application, we obtain that general complete hypertetrahedral arrangements are not locally free. In the second part of this paper, we bound the initial degree of the first syzygy module of the Jacobian ideal of $\mathcal{A}$., Comment: 26 pages. To appear in Mathematische Zeitschrift. Comments are welcome

Let $R$ be a Noetherian local ring. We prove that $R$ is regular of dimension at most four if, and only if, every prime ideal, defining a Gorenstein quotient ring, is syzygetic. We deduce a characterization of these rings in terms of the Andr\'e-Quillen homology., Comment: To appear in Journal of Algebra

"Vegeu el resum a l'inici del documents del fitxer adjunt".

The goal of this paper is to explicitly describe a minimal binomial generating set of a class of lattice ideals, namely the ideal of certain Veronese 3-fold projections. More precisely, for any integer d ≥ 4 and any d -th root e of 1 we denote by X d the toric variety defined as the image of the morphism φ T d : P 3 ⟶ P μ ( T d ) − 1 where T d are all monomials of degree d in k [ x , y , z , t ] invariant under the action of the diagonal matrix M ( 1 , e , e 2 , e 3 ) . In this work, we describe a Z -basis of the lattice L η associated to I ( X d ) as well as a minimal binomial set of generators of the lattice ideal I ( X d ) = I + ( η ) .

This paper investigates the existence and properties of a Bernstein-Sato functional equation in nonregular settings. In particular, we construct $D$-modules in which such formal equations can be studied. The existence of the Bernstein-Sato polynomial for a direct summand of a polynomial over a field is proved in this context. It is observed that this polynomial can have zero as a root, or even positive roots. Moreover, a theory of $V$-filtrations is introduced for nonregular rings, and the existence of these objects is established for what we call differentially extensible summands. This family of rings includes toric, determinantal, and other invariant rings. This new theory is applied to the study of multiplier ideals and Hodge ideals of singular varieties. Finally, we extend known relations among the objects of interest in the smooth case to the setting of singular direct summands of polynomial rings., 42 pages. A new section on Hodge ideals is included. Comments welcome

In this work we introduce a new set of invariants associated to the linear strands of a minimal free resolution of a $\mathbb{Z}$-graded ideal $I\subseteq R=\Bbbk[x_1, \ldots, x_n]$. We also prove that these invariants satisfy some properties analogous to those of Lyubeznik numbers of local rings. In particular, they satisfy a consecutiveness property that we prove first for Lyubeznik numbers. For the case of squarefree monomial ideals we get more insight on the relation between Lyubeznik numbers and the linear strands of their associated Alexander dual ideals. Finally, we prove that Lyubeznik numbers of Stanley-Reisner rings are not only an algebraic invariant but also a topological invariant, meaning that they depend on the homeomorphic class of the geometric realization of the associated simplicial complex and the characteristic of the base field., Comment: 25 pages. Accepted in Nagoya Math. J

We introduce a formalism to produce several families of spectral sequences involving the derived functors of the limit and colimit functors over a finite partially ordered set. The first type of spectral sequences involves the left derived functors of the colimit of the direct system that we obtain applying a family of functors to a single module. For the second type we follow a completely different strategy as we start with the inverse system that we obtain by applying a covariant functor to an inverse system. The spectral sequences involve the right derived functors of the corresponding limit. We also have a version for contravariant functors. In all the introduced spectral sequences we provide sufficient conditions to ensure their degeneration at their second page. As a consequence we obtain some decomposition theorems that greatly generalize the well-known decomposition formula for local cohomology modules given by Hochster., Comment: 63 pages, comments are welcome. To appear in International Mathematics Research Notices

We introduce the notion of relation type of an affine algebra and prove that it is well defined by using the Jacobi-Zariski exact sequence of Andre-Quillen homology. In particular, the relation type,is an invariant of an affine algebraic variety. Also as a consequence of the invariance, we show that in order to calculate the relation type of an ideal in a polynomial ring one can reduce the problem to trinomial ideals. When the relation type is at least two, the extreme equidimensional components play no role. This leads to the non-existence of affine algebras of embedding dimension three and relation type two. (C) 2015 Elsevier Inc. All rights reserved.

We give formulas and effective sharp bounds for the degree of multi-graded rational maps and provide some effective and computable criteria for birationality in terms of their algebraic and geometric properties. We also extend the Jacobian dual criterion to the multi-graded setting. Our approach is based on the study of blow-up algebras, including syzygies, of the ideal generated by the defining polynomials of the rational map. A key ingredient is a new algebra that we call the "saturated special fiber ring", which turns out to be a fundamental tool to analyze the degree of a rational map. We also provide a very effective birationality criterion and a complete description of the equations of the associated Rees algebra of a particular class of plane rational maps., Comment: to appear in Proc. Lond. Math. Soc

In this paper we work with several divisors of a module E ⊆ G ≃ R e having rank e , such as the classical Fitting ideals of E and of G / E , and the more recently introduced (generic) Bourbaki ideals I ( E ) (Simis et al. (2003) [19] ) or ideal norms [ [ E ] ] R (Villamayor (2006) [23] ). We found several relations and equalities among them which allow to describe in some cases universal properties with respect to E of their blow ups. As a byproduct we are also able to obtain lower bounds for the analytic spread l ( ⋀ e E ) , related with the algebraic local version of Zak’s inequality as explained in Simis et al. (2002) [17] .

We describe geometrically and algebraically the set of unattainable points for the Rational Hermite Interpolation Problem (i.e. those points where the problem does not have a solution). We show that this set is a union of equidimensional complete intersection varieties of odd codimension, the number of them being equal to the minimum between the degrees of the numerator and denominator of the problem. Each of these equidimensional varieties can be further decomposed as a union of as many rational (irreducible) varieties as input data points. We exhibit algorithms and equations defining all these objects., Comment: 22 pages, amsart. Revised version accepted for publication in Linear Algebra and its Applications

Treballs finals del Màster en Matemàtica Avançada, Facultat de matemàtiques, Universitat de Barcelona, Any: 2017, Director: Santiago Zarzuela, [en] Let $\Delta$ be a triangulation of a $(d - 1)$-dimensional sphere with $n$ vertices. The Upper Bound Conjecture (UBC for short) gives an explicit bound of the number of $i$-dimensional faces of $\Delta$. This question dates back to the beginning of the 1950’s, when the study of the efficiency of some linear programming techniques led to the following problem: Determine the maximal possible number of $i$-faces of d-polytope with $n$ vertices. The first statement of the UBC was formulated in 1957 by Theodore Motzkin. The original result state that the number of $i$-dimensional faces of a $d$-dimensional polytope with n vertices are bound by a certain explicit number $f i (C(n, d))$ where $C(n, d)$ is a cyclic polytope and $f_{i}$ denotes the number of $i$-dimensional faces of the simplex. We say that $P$ is a polytope if it is the convex hull of a finite set of points in $\mathbb{R}^{d}$. Moreover, we say that $C(n, d)$ is a cyclic polytope if it is the convex hull of n distinct points on the moment curve $(t, t^{2},..., t{^d})$, $-\infty

Treballs finals del Màster en Matemàtica Avançada, Facultat de matemàtiques, Universitat de Barcelona, Any: 2016, Director: Carles Casacuberta, The central topic of this work is the concept of acyclic spaces in topological K-theory and their analogues in algebraic K-theory. We start by describing topological K-theory and some basic results, such as representability by a spectrum. Next we discuss algebraic K-theory and some of its properties, including Swan’s theorem, followed by the topological tools required to construct higher algebraic K-theory by means of Quillen’s plus-construction. Finally, we describe a class of rings whose algebraic K-theory groups vanish in all dimensions. In fact each ring $R$ admits a cone $CR$ with $K_i (CR) = 0$ for all i and a suspension $SR$ that is used to define negative K-theory groups of R in analogy with the topological case.

In this paper, we use the Ap\'ery table of the numerical semigroup associated to an affine monomial curve in order to characterize arithmetic properties and invariants of its tangent cone. In particular, we precise the shape of the Ap\'ery table of a numerical semigroup of embedding dimension 3, when the tangent cone of its monomial curve is Buchsbaum or 2-Buchsbaum, and give new proofs for two conjectures raised by V. Sapko (Commun. Algebra {29}:4759-4773, 2001) and Y. H. Shen (Commun. Algebra {39}:1922-1940, 2001). We also provide a new simple proof in the case of monomial curves for Sally's conjecture (Numbers of Generators of Ideals in Local Rings, 1978) that the Hilbert function of a one-dimensional Cohen-Macaulay ring with embedding dimension three is non-decreasing. Finally, we obtain that monomial curves of embedding dimension 4 whose tangent cones are Buchsbaum, and also monomial curves of any embedding dimensions whose numerical semigroups are balanced, have non-decreasing Hilbert functions. Numerous examples are provided to illustrate the results, most of them computed by using the NumericalSgps package of GAP (Delgado et al., NumericalSgps-a GAP package, 2006)., Comment: To appear in Semigroup Forum

This paper takes a new look at ideals generated by 2×2 minors of 2×3 matrices whose entries are powers of three elements not necessarily forming a regular sequence. A special case of this is the ideals determining monomial curves in three-dimensional space, which were studied by Herzog. In the broader context studied here, these ideals are identified as Northcott ideals in the sense of Vasconcelos, and so their liaison properties are displayed. It is shown that they are set-theoretically complete intersections, revisiting the work of Bresinsky and of Valla. Even when the three elements are taken to be variables in a polynomial ring in three variables over a field, this point of view gives a larger class of ideals than just the defining ideals of monomial curves. We then characterize when the ideals in this larger class are prime, we show that they are usually radical and, using the theory of multiplicities, we give upper bounds on the number of their minimal prime ideals, one of these primes being a uniquely determined prime ideal of definition of a monomial curve. Finally, we provide examples of characteristic-dependent minimal prime and primary structures for these ideals.

In this note we compute the Poincaré series of almost stretched Gorenstein local rings. It turns out that it is rational.

We prove that sequentially Cohen-Macaulay rings in positive characteristic, as well as sequentially Cohen-Macaulay Stanley-Reisner rings in any characteristic, have trivial Lyubeznik table. Some other configurations of Lyubeznik tables are also provided depending on the deficiency modules of the ring.

Treballs finals del Màster en Matemàtica Avançada, Facultat de matemàtiques, Universitat de Barcelona, Any: 2014, Director: Joan Elías García, The aim of this work is to present some recent results on the classification of Artin Gorenstein by using the inverse system of Macaulay. Furthermore, the study of the inverse system will alow to introduce the concepts of injective modules, the injective envelope or the Matlis duality. But a good question would be that why is important to study the Artinian algebras.

We prove that the Frobenius algebra of the injective hull of a complete Stanley-Reisner ring as well as its Matlis dual notion of Cartier algebra can be only principally generated or infinitely generated. As a consequence we are able to show that the set of F-jumping numbers of generalized test ideals associated to complete Stanley-Reisner rings form a discrete set., 16 pages

In this paper, we investigate galois theory of CP-graded ring extensions. In particular, we generalize some galois results given in [1, 2] and, without restriction to nor graded fields nor torsion free of the grade groups, we show that some results of graded field extensions given in [3] hold.

[eng] The aim of this work is to study the ring-theoretic properties of the diagonals of a Rees algebra, w that we shall review the known results about these problems, and finally we will give a summary of the contents and results obtained in this work.hich from a geometric point of view are the homogenous coordinate rings of embeddings of blow-ups of projective varieties along a subvariety. First we are going to introduce the subject and the main problems. After that we shall review the known results about these problems, and finally we will give a summary of the contents and results obtained in this work., [cat] L’objectiu d’aquesta memòria és l’estudi de les propietats aritmètiques de les diagonals d’una àlgebra de Rees o, des d’un punt de vista geomètric, dels anells de coordenades homogenis d’immersions d’explosions de varietats projectives al llarg d’una subvarietat. En primer lloc, anem a introduir el tema i els principals problemes que tractarem. A continuació, exposarem els resultats coneguts sobre aquests problemes i finalment farem un resum dels resultats obtinguts en aquesta memòria.

Let Y be a closed subscheme of Pn−1 k defined by a homogeneous ideal I⊂ A=k[X1,...,Xn], and X obtained by blowing up Pn−1 k along Y. Denote by Ic the degree c part of I and assume that I is generated by forms of degree ≤ d. Then the rings k[(Ie)c] are coordinate rings of projective embeddings of X in PN−1 k , where N=dimk(Ie)c for c ≥ de+1. The aim of this paper is to study the Gorenstein property of the rings k[(Ie)c] . Under mild hypothesis we prove that there exist at most a finite number of diagonals (c, e) such that k[(Ie)c] is Gorenstein, and we determine them for several families of ideals.

A maximal disjoint subset $S$ of an $MV$-algebra $A$ is a basis iff $\{x \in A : x \leq a \}$ is a linearly ordered subset of $A$ for all $a \in S$. Let $\Spec A$ be the set of the prime ideals of $A$ with the usual spectral topology. A decomposition $\Spec A = \cup_{i \in I} T_{i} \cup X$ is said to be orthogonal iff each $T_{i}$ is compact open and $S = \{a_{i}\}_{i\in I}$ is a maximal disjoint subset. We prove that this decomposition is unrefinable (i.e. no $T_{i} = \Theta \cap Y$ with $\Theta$ open, $\Theta \cap Y = \emptyset$, int $Y = \emptyset$) iff $S$ is a basis. Many results are established for semisimple $MV$-algebras, which are the algebraic counterpart of Bold fuzzy set theory.

Preprint enviat per a la seva publicació: Capítol del llibre: Commutative algebra, Algebraic geometry, and computational methods, 1999, p. 215-222

$MV$-algebras were introduced in 1958 by Chang and they are models of Lukasiewicz infinite-valued logic. Chang gives a correspondence between the category of linearly ordered $MV$-algebras and the category of linearly ordered abelian $\ell$-groups. Mundici extended this result showing a categorical equivalence between the category of the $MV$-algebras and the category of the abelian $\ell$-groups with strong unit. In this paper, starting from some definitions and results in abelian $\ell$-groups, we shall study the convergent sequences and the Cauchy sequences in an $MV$-algebra. The main result is the construction of the Cauchy completion $A^{*}$ of an $MV$-algebra $A$. It is proved that a complete $MV$-algebra is also Cauchy complete. Additional results on atomic and complete $MV$-algebras are also given.

We introduce the notion of {\it p-ideal\/} of a QMV-algebra and we prove that the class of all $p$-ideals of a QMV-algebra $\C M$ is in one-to-one correspondence with the class of all congruence relations of $\C M$.

In this paper we define maximal $MV$-algebras, a concept similar to the maximal rings and maximal distributive lattices. We prove that any maximal $MV$-algebra is semilocal, then we characterize a maximal $MV$-algebras as finite direct product of local maximal $MV$-algebras.

Preprint enviat per a la seva publicació en una revista científica: Journal of Algebra. 15 December 1997, Volume 198, Issue 2, Pages 428-445. [https://doi.org/10.1006/jabr.1997.7147]

Let $I$ be an ideal of a commutative Noetherian ring $A$, $A\supset \Q$, $B=A/I$ and ${\bf G}(I)$ the associated graded ring to $I$. It is known that $H_{2}(A,B,B)=0$ is equivalent to $I$ being syzygetic. We prove that the vanishing of $\hd $ is equivalent to $I$ being of linear type and $\sigma _{3,q}:\extp{3}{B}{I/I^{2}}\otimes _{B}I^{q}/I^{q+1}\rightarrow \tor{3}{A}{B}{A/I^{q+1}}$, the $(3,q)$-antisymmetrization morphism, being surjective for all $q\geq 0$. Using this and a theorem of Ulrich on a conjecture of Herzog, we deduce that, in a regular local ring $A$, a Gorenstein, licci ideal $I$ verifies $\hd =0$ if and only if $I$ is a complete intersection. Thus, we characterize perfect (respectively, Gorenstein) ideals of grade two (respectively, three) with $\hd =0$ as those ideals which are of linear type (respectively, complete intersection). With any grade, but small deviation, we show that a licci ideal, generically a complete intersection and of deviation one, verifies $\hd =0$. This is not true for licci ideals of linear type and of deviation two.

Preprint enviat per a la seva publicació en una revista científica: Manuscripta Mathematica, 1998, vol. 95, pp. 47–58. [https://doi.org/10.1007/BF02678014], We consider the blowing up of ℙ k/n−1 along a closed subscheme defined by a homogeneous idealI ∪A=k[X 1, …,X n ] generated by forms of degree ≤d, and its projective embeddings by the linear systems corresponding to (I e) c , forc≥de+1. The homogeneous coordinate rings of these embeddings arek[(I e) c ]. One wants to study the Cohen-Macaulay property of these rings. We will prove that if the Rees algebraR A (I) is Cohen-Macaulay, thenk[(I e) c ] are Cohen-Macaulay forc>>e>0, thus proving a conjecture stated by A. Conca, J. Herzog, N.V. Trung and G. Valla.