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CAPES - Coordenação de Aperfeiçoamento de Pessoal de Nível Superior O principal objetivo desta dissertação é estudar os ideais injetivos e sobrejetivos de polinômios homogêneos entre espaços de Banach. Primeiramente faremos uma breve exposição sobre polinômios homogêneos e ideais de polinômios homogêneos. Em seguida estudamos os espaços $\ell_\infty(\Gamma)$ e $\ell_1(\Gamma)$, que reúnem famílias escalares indexadas em um conjunto arbitrário não vazio $\Gamma$, munidos da norma do supremo e da norma da soma, respectivamente, assim como as injeções e sobrejeções métricas e as propriedades de extensão e levantamento desses espaços. Posteriormente estudamos os ideais injetivos de polinômios, com ênfase na envoltória injetiva e na descrição da envoltória injetiva de um ideal de composição. Finalmente fazemos um estudo análogo para o caso de ideais sobrejetivos de polinômios. Aplicações para as descrições das envoltórias injetiva e sobrejetiva dos ideais de composição são apresentadas. The main purpose of this dissertation is to study injective and surjective ideals of homogeneous polynomials between Banach spaces. First we briefly sketch the theory of homogeneous polynomials and polynomial ideals. Next we study the spaces $\ell_\infty(\Gamma)$ and $\ell_1(\Gamma)$, formed by scalar families indexed by an arbitrary non void set $\Gamma$ endowed with the supremum norm and the sum norm, respectively, as well as metric injections and surjections and extension and lifting properties of such spaces. Then, injective polynomials ideals are investigated, with a special attention to the injective hull and to the description of the injective hull of a composition ideal. Finally we undertake a similar study concerning surjective polynomial ideals. Applications of the descriptions of the injective and surjective hulls of composition ideals are provided. Dissertação (Mestrado)

Coordenação de Aperfeiçoamento de Pessoal de Nível Superior The main purpose of this dissertation is the study of ideals of multilinear mappings and homogeneous polynomials between linear spaces. By an ideal we mean a class that is stable under the composition with linear operators. First we study multilinear mappings and spaces of multilinear mappings. We also show how to obtain, from a given multilinear mapping, other multilinear mappings with degrees of multilinearity greater than, equal to or smaller than the degree of the original multilinear mapping. Next we study homogeneous polynomials and spaces of homogeneous polynomials, and we also show how to obtain, from a given n-homogeneous polynomial, other polynomials with degrees of homogeneity greater than, equal to or smaller than the degree of the original polynomial. Next we study ideals of multilinear mappings, or multi-ideals, and ideals of homogeneous polynomial, or polynomial ideals, giving several examples and presenting methods to generated multi-ideals and polynomial ideals from a given operator ideal. Finally we dene and give several examples of coherent multi-ideals and coherent polynomial ideals. O principal objetivo desta dissertação e estudar os ideais de aplicações multilineares e polinômios homogêneos entre espaços vetoriais. Por um ideal entendemos uma classe de aplicações que e estavel atraves da composição com operadores lineares. Primeiramente estudamos as aplicações multilineares e os espaços de aplicações multilineares. Mostramos tambem como obter, a partir de uma aplicação multilinear dada, outras aplicações com graus de multilinearidade maiores, iguais ou menores que o da aplicação original. Em seguida estudamos os polinômios homogêneos e os espacos de polinômios homogêneos, e mostramos que, a partir de um polinômio n-homogêneo, tambem podemos construir novos polinômios homogêneos com graus de homogeneidade maiores, iguais ou menores que n. Posteriormente estudamos os ideais de aplicações multilineares, ou multi-ideais, e os ideais de polinômios homogêneos, exibindo varios exemplos e apresentando metodos para se obter um multi-ideais, ou ideais de polinômios, a partir de ideais de operadores lineares dados. Por m, denimos e exibimos varios exemplos de multi-ideais coerentes e de ideais coerentes de polinômios. Mestre em Matemática

In this paper, we study the internal stabilizability and internal stabilization problems for multidimensional (nD) systems. Within the fractional representation approach, a multidimen-sional system can be studied by means of matrices with entries in the integral domain of structurally stable rational fractions, namely the ring of rational functions which have no poles in the closed unit polydisc U n = {z = (z 1 ,. .. , z n) $\in$ C n | |z 1 | 1,. .. , |z n | 1}. It is known that the internal stabilizability of a multidimensional system can be investigated by studying a certain polynomial ideal I = p 1 ,. .. , p r that can be explicitly described in terms of the transfer matrix of the plant. More precisely the system is stabilizable if and only if V (I) = {z $\in$ C n | p 1 (z) = $\times$ $\times$ $\times$ = p r (z) = 0} $\cap$ U n = $\emptyset$. In the present article, we consider the specific class of linear nD systems (which includes the class of 2D systems) for which the ideal I is zero-dimensional, i.e., the p i 's have only a finite number of common complex zeros. We propose effective symbolic-numeric algorithms for testing if V (I) $\cap$ U n = $\emptyset$, as well as for computing, if it exists, a stable polynomial p $\in$ I which allows the effective computation of a stabilizing controller. We illustrate our algorithms through an example and finally provide running times of prototype implementations for 2D and 3D systems.

2018 International Symposium on Symbolic and Algebraic Computation (ISSAC), July 2018, New York, NY, United States, We present a deterministic algorithm for deciding if a polynomial ideal, with coefficients in an algebraically closed field K of characteristic zero, of which we know just some very limited data, namely:the number n of variables, and some upper bound for the geometric degree of its zero set in Kn, is or not the zero ideal. The algorithm performs just a finite number of decisions to check whether a point is or not in the zero set of the ideal. Moreover, we extend this technique to test, in the same fashion, if the elimination of some variables in the given ideal yields or not the zero ideal. Finally, the role of this technique in the context of automated theorem proving of elementary geometry statements, is presented, with references to recent documents describing the excellent performance of the already existing prototype version, implemented in GeoGebra., Agencia Estatal de Investigación

Source localization and spectral estimation are among the most fundamental problems in statistical and array signal processing. Methods which rely on the orthogonality of the signal and noise subspaces, such as Pisarenko's method, MUSIC, and root-MUSIC are some of the most widely used algorithms to solve these problems. As a common feature, these methods require both apriori knowledge of the number of sources, and an estimate of the noise subspace. Both requirements are complicating factors to the practical implementation of the algorithms, and when not satisfied exactly, can potentially lead to severe errors. In this paper, we propose a new localization criterion based on the algebraic structure of the noise subspace that is described for the first time to the best of our knowledge. Using this criterion and the relationship between the source localization problem and the problem of computing the greatest common divisor (GCD), or more practically approximate GCD, for polynomials, we propose two algorithms which adaptively learn the number of sources and estimate their locations. Simulation results show a significant improvement over root-MUSIC in challenging scenarios such as closely located sources, both in terms of detection of the number of sources and their localization over a broad and practical range of SNRs. Further, no performance sacrifice in simple scenarios is observed., Comment: 16 two-column pages, 12 figures, 1 table, Submitted for publication in IEEE Trans. Signal Processing

2018 International Symposium on Symbolic and Algebraic Computation (ISSAC), July 2018, New York, NY, United States We present a deterministic algorithm for deciding if a polynomial ideal, with coefficients in an algebraically closed field K of characteristic zero, of which we know just some very limited data, namely:the number n of variables, and some upper bound for the geometric degree of its zero set in Kn, is or not the zero ideal. The algorithm performs just a finite number of decisions to check whether a point is or not in the zero set of the ideal. Moreover, we extend this technique to test, in the same fashion, if the elimination of some variables in the given ideal yields or not the zero ideal. Finally, the role of this technique in the context of automated theorem proving of elementary geometry statements, is presented, with references to recent documents describing the excellent performance of the already existing prototype version, implemented in GeoGebra. Agencia Estatal de Investigación

We provide coincidence results for vector-valued ideals of multilinear operators. More precisely, if A is an ideal of n-linear mappings we give conditions for which the equality A(E1, ..., En; F) = Amin(E1, ..., En; F) holds isometrically. As an application, we obtain in many cases that the monomials form a Schauder basis of the space A(E1, ..., En; F). Several structural and geometric properties are also derived using this equality. We apply our results to the particular case where A is the classical ideal of extendible or Pietsch-integral multilinear operators. Similar statements are given for ideals of vector-valued homogeneous polynomials. Fil: Galicer, Daniel Eric. Consejo Nacional de Investigaciones Científicas y Técnicas. Oficina de Coordinación Administrativa Ciudad Universitaria. Instituto de Investigaciones Matemáticas "Luis A. Santalo". Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales. Instituto de Investigaciones Matemáticas "Luis A. Santalo"; Argentina Fil: Villafañe, Norberto Román. Consejo Nacional de Investigaciones Científicas y Técnicas. Oficina de Coordinación Administrativa Ciudad Universitaria. Instituto de Investigaciones Matemáticas "Luis A. Santalo". Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales. Instituto de Investigaciones Matemáticas "Luis A. Santalo"; Argentina

A wealth of geometric and combinatorial properties of a given linear endomorphism $X$ of $\R^N$ is captured in the study of its associated zonotope $Z(X)$, and, by duality, its associated hyperplane arrangement ${\cal H}(X)$. This well-known line of study is particularly interesting in case $n\eqbd\rank X \ll N$. We enhance this study to an algebraic level, and associate $X$ with three algebraic structures, referred herein as {\it external, central, and internal.} Each algebraic structure is given in terms of a pair of homogeneous polynomial ideals in $n$ variables that are dual to each other: one encodes properties of the arrangement ${\cal H}(X)$, while the other encodes by duality properties of the zonotope $Z(X)$. The algebraic structures are defined purely in terms of the combinatorial structure of $X$, but are subsequently proved to be equally obtainable by applying suitable algebro-analytic operations to either of $Z(X)$ or ${\cal H}(X)$. The theory is universal in the sense that it requires no assumptions on the map $X$ (the only exception being that the algebro-analytic operations on $Z(X)$ yield sought-for results only in case $X$ is unimodular), and provides new tools that can be used in enumerative combinatorics, graph theory, representation theory, polytope geometry, and approximation theory., Comment: 44 pages; updated to reflect referees' remarks and the developments in the area since the article first appeared on the arXiv

We show that for each Banach ideal of homogeneous polynomials, there exists a (necessarily unique) Banach operator ideal compatible with it. Analogously, we prove that any ideal of $n$-homogeneous polynomials belongs to a coherent sequence of ideals of $k$-homogeneous polynomials., Comment: 12 pages

We study Poincare duality algebras over the field F 2 of two elements. After introducing a connected sum operation for such algebras we compute the corresponding Grothendieck group of surface algebras (i.e., Poincare algebras of formal dimension 2). We show that the corresponding group for 3-folds (i.e., algebras of formal dimension 3) is not finitely generated, but does have a Krull–Schmidt property. We then examine the isomorphism classes of 3-folds with at most three generators of degree 3, provide a complete classification, settle which such occur as the cohomology of a smooth 3-manifold, and list separating invariants. The closing section and Appendix A provide several different means of constructing connected sum indecomposable 3-folds.

In this paper we show that the ideal of any algebraic curve in affine 3-space whose Jacobian matrix has rank at least 1 at every singular point of the curve can be generated by three polynomials and we give constructive procedures to compute such generators.

Let $A$ be a subvariety of affine space $\mathbb{A}^n$ whose irreducible components are $d$-dimensional linear or affine subspaces of $\mathbb{A}^n$. Denote by $D(A)\subset\mathbb{N}^n$ the set of exponents of standard monomials of $A$. We show that the combinatorial object $D(A)$ reflects the geometry of $A$ in a very direct way. More precisely, we define a $d$-plane in $\mathbb{N}^n$ as being a set $\gamma+\oplus_{j\in J}\mathbb{N}e_{j}$, where $#J=d$ and $\gamma_{j}=0$ for all $j\in J$. We call the $d$-plane thus defined to be parallel to $\oplus_{j\in J}\mathbb{N}e_{j}$. We show that the number of $d$-planes in $D(A)$ equals the number of components of $A$. This generalises a classical result, the finiteness algorithm, which holds in the case $d=0$. In addition to that, we determine the number of all $d$-planes in $D(A)$ parallel to $\oplus_{j\in J}\mathbb{N}e_{j}$, for all $J$. Furthermore, we describe $D(A)$ in terms of the standard sets of the intersections $A\cap\{X_{1}=\lambda\}$, where $\lambda$ runs through $\mathbb{A}^1$., Comment: 31 pages, 8 figures

We study the existence of atomic decompositions for tensor products of Banach spaces and spaces of homogeneous polynomials. If a Banach space X admits an atomic decomposition of a certain kind, we show that the symmetrized tensor product of the elements of the atomic decomposition provides an atomic decomposition for the symmetric tensor product ⊗s, μ n X, for any symmetric tensor norm μ. In addition, the reciprocal statement is investigated and analogous consequences for the full tensor product are obtained. Finally we apply the previous results to establish the existence of monomial atomic decompositions for certain ideals of polynomials on X. © 2008 Elsevier Inc. All rights reserved. Fil:Carando, D. Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales; Argentina. Fil:Lassalle, S. Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales; Argentina.

We present a static analysis that approximates the relational semantics of imperative programs by systems of low-degree polynomial equalities. Our method is based on Abstract Interpretation in a lattice of polynomial pseudo ideals — finite-dimensional vector spaces of degree-bounded polynomials that are closed under degree-bounded products. For a fixed degree bound, the sizes of bases of pseudo ideals and the lengths of chains in the lattice of pseudo ideals are bounded by polynomials in the number of program variables. Despite the approximate nature of our analysis, for several programs taken from the literature on non-linear polynomial invariant generation our method produces results that are as precise as those produced by methods based on polynomial ideals and Gröbner bases.

Esta tesis tiene como objeto contribuir al desarrollo de la teoría métrica e isomorfa de productos tensoriales simétricos en espacios de Banach. Mostramos varios ejemplos donde la teoría de ideales de polinomios homogéneos resulta enriquecida con el uso de técnicas tensoriales. Probamos que la extensión de Aron-Berner preserva la norma para todo ideal maximal y minimal de polinomios homogéneos. Este resultado puede interpretarse como una versión polinomial de uno de los “Cinco Lemas Básicos” de la teoría de productos tensoriales. Más aún, enunciamos y probamos análogos simétricos de dichos lemas y damos, a lo largo del texto, varias aplicaciones. Estudiamos las cápsulas inyectivas y projectivas de una norma tensorial simétrica, analizando sus propiedades y relaciones. Describimos los ideales de polinomios maximales asociados a dichas normas en términos de ideales de composición e ideales cocientes. Examinamos las normas naturales de Grothendieck en el n-ésimo producto tensorial simétrico y mostramos que, para n ≥ 3, hay exactamente seis de ellas, a diferencia del caso n = 2 donde hay cuatro. Definimos la propiedad de Radón-Nikodym simétrica para normas s-tensoriales y mostramos, bajo ciertas hipótesis, que los ideales de polinomios maximales asociados a normas con dicha propiedad coinciden isométricamente con su núcleo minimal. Como consecuencia, probamos la existencia de ciertas estructuras en algunos ideales de polinomios clásicos (existencia de bases o la propiedad de Radon-Nikodym). Por otra parte, damos una demostración alternativa del hecho que el ideal de los polinomios integrales coincide isométricamente con el ideal de los polinomios nucleares en espacios Asplund. Analizamos la existencia de bases incondicionales en ideales de polinomios. Para esto, estudiamos incondicionalidad en productos tensoriales simétricos. Damos un criterio sencillo para determinar si un ideal de polinomios carece de base incondicional. Utilizando dicho criterio mostramos que muchos de los ideales usuales no poseen estructura incondicional. Entre ellos, los r-integrales, r-dominados, extendibles y r-factorizables. Para muchos de estos ejemplos obtenemos incluso que la sucesión básica monomial no es incondicional. Estudiamos la preservación de otro tipo de estructuras en el producto tensorial simétrico: la estructura de álgebra de Banach y la estructura de M-ideal. Mostramos cuáles de las normas s-tensoriales de Grothendieck preservan la estructura de álgebra. Por otra parte, probamos que la norma inyectiva simétrica destruye la estructura de M-ideal (opuesto a lo que pasa en el producto tensorial completo con la norma inyectiva). Si bien dicha estructura se pierde en el caso simétrico, mostramos que, si E es Asplund y M-ideal en F, entonces los polinomios integrales sobre E se extienden a F preservando la norma de manera única. This thesis aims to contribute to the development of the metric and isomorphic theory of symmetric tensor products on Banach spaces. We show several examples where the theory of polynomial ideals is enriched with the use of tensor techniques. We prove that the Aron-Berner extension preserves the norm for every maximal and minimal ideal of homogeneous polynomials. This result can be interpretated as a polynomial version of one of the “Five basic Lemmas” of the theory of tensor products. Moreover, we state and prove symmetric analogues of these lemmas and give, throughout the text, several applications. We study the injective and projective associates of a symmetric tensor norm, analyzing their properties and relations. We describe the maximal polynomial ideals associated with these norms in terms of composition ideals and quotient ideals. We examine Grothendieck’s natural norms on the n-fold symmetric tensor product and show that there are exactly six natural symmetric tensor norms for n ≥ 3, unlike the 2-fold case in which there are four. We define the symmetric Rad´on-Nikod´ym property for s-tensor norms and show, under certain hypothesis, that maximal polynomial ideals associated with norms with this property coincide isometrically with their minimal kernel. As a consequence, we prove the existence of certain structures on some classical polynomial ideals (existence of basis or the Rad´on- Nikod´ym property). On the other hand, we give an alternative proof of the fact that the ideal of integral polynomials coincide isometrically with the ideal of nuclear polynomials on Asplund spaces. We analyze the existence of unconditional basis on polynomial ideals. For this, we study unconditionality on symmetric tensor products. We provide a simple criterium to check wether a polynomial ideal lacks of unconditional basis. Using this criterium, we show that many usual polynomial ideals do not have unconditional structure. Among them we have the r-integral, r-dominated, extendible and r-factorable polynomials. For many of these examples we also get that the monomial basic sequence is not unconditional. We study the preservation of other kind of structures on the symmetric tensor product: the Banach algebra structure and the M-ideal structure. We show which of the Grothendieck’s natural symmetric tensor norms preserve the algebra structure. On the other hand, we prove that the injective s-tensor norm destroys theM-ideal structure (opposite to what happens in the full tensor product with the injective norm). Even though this structure is lost in the symmetric case, we show that, if E is Asplund and M-ideal in F, every integral polynomial in E has a unique norm preserving extension to F. Fil: Galicer, Daniel E.. Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales; Argentina.

En esta tesis nos enfocamos en los aspectos algorítmicos de algunos de los tópicos más importantes del álgebra conmutativa. Estudiamos el cálculo de radicales y primos y minimales, la normalización de anillos e ideales y otros problemas relacionados. En los últimos años, se desarrollaron varios programas de álgebra computacional con implementaciones muy eficientes de las herramientas básicas para trabajar con polinomios, ideales y anillos. Esto renovó el interés por algoritmos eficientes para resolver algunos problemas difíciles del área. Proponemos nuevos algoritmos para algunos de estos problemas, basándonos en ideas matemáticas y resultados nuevos. Hemos implementado todos los algoritmos en esta tesis en Singular (Decker et al., 2011), uno de los programas de álgebra computacional más comúnmente utilizados, y están actualmente disponibles para su uso por toda la comunidad matemática. Si bien para la mayoría de estos problemas ya existían algoritmos, los nuevos algoritmos propuestos los superan en la mayoría de los casos, siendo ahora los algoritmos por default en SINGULAR. This thesis addresses the algorithmic aspects of some major topics of commutative algebra. We study the computation of radicals and minimal associated primes of ideals, the normalization of rings and ideals and other related problems. In recent years a number of computer algebra systems have been developed with very efficient implementations of some basic tools to work with polynomials, ideals and rings. This put on the spot the need for efficient algorithms to solve some difficult problems. We propose new algorithms for some of these problems, based on new mathematical ideas and results. All the algorithms in this thesis have been implemented in Singular (Decker et al., 2011), one of the most commonly used computer algebra systems, and are now available for use of the mathematical community. Although other algorithms already existed for most of these tasks, the new algorithms outperform them in most cases and are now the default algorithms in SINGULAR. Fil: Laplagne, Santiago Jorge. Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales; Argentina.

We show that several spaces of holomorphic functions on a Riemann domain over a Banach space, including the nuclear and Hilbert-Schmidt bounded type, are locally $m$-convex Fr\'echet algebras. We prove that the spectrum of these algebras has a natural analytic structure, which we use to characterize the envelope of holomorphy. We also show a Cartan-Thullen type theorem., Comment: 30 pages

We introduce the symmetric-Radon-Nikod\'ym property (sRN property) for finitely generated s-tensor norms $\beta$ of order $n$ and prove a Lewis type theorem for s-tensor norms with this property. As a consequence, if $\beta$ is a projective s-tensor norm with the sRN property, then for every Asplund space $E$, the canonical map $\widetilde{\otimes}_{\beta}^{n,s} E' \to \Big(\widetilde{\otimes}_{\beta'}^{n,s} E \Big)'$ is a metric surjection. This can be rephrased as the isometric isomorphism $\mathcal{Q}^{min}(E) = \mathcal{Q}(E)$ for certain polynomial ideal $\Q$. We also relate the sRN property of an s-tensor norm with the Asplund or Radon-Nikod\'{y}m properties of different tensor products. Similar results for full tensor products are also given. As an application, results concerning the ideal of $n$-homogeneous extendible polynomials are obtained, as well as a new proof of the well known isometric isomorphism between nuclear and integral polynomials on Asplund spaces., Comment: 17 pages

Given an homogeneous polynomial on a Banach space $E$ belonging to some maximal or minimal polynomial ideal, we consider its iterated extension to an ultrapower of $E$ and prove that this extension remains in the ideal and has the same ideal norm. As a consequence, we show that the Aron-Berner extension is a well defined isometry for any maximal or minimal ideal of homogeneous polynomials. This allow us to obtain symmetric versions of some basic results of the metric theory of tensor products., Comment: 13 pages

Definimos el concepto de sucesión coherente de ideales de polinomios en espacios de Banach, que nos permite relacionar ideales de polinomios homogéneos de diferentes grados. A cada sucesión coherente A, podemos asociarle un espacio de Fréchet de funciones enteras de tipo acotado, HbA. Extendemos a HbA un resultado de Godefroy y Shapiro sobre hiperciclicidad de operadores de convolución. También estudiamos el concepto de sucesión multiplicativa de ideales de polinomios a valores escalares. Esto nos permite asociar un álgebra de funciones enteras de tipo acotado HbA a cada sucesión coherente y multiplicativa de ideales de polinomios, A. Probamos que, bajo ciertas condiciones naturales, el espectro del álgebra asociada, MbA, puede ser dotado de una estructura de dominio de Riemann sobre el bidual del espacio de Banach. Además la extensión de cada función de HbA al espectro es una función A-holomorfa de tipo acotado en cada componente conexa. Investigamos cómo definir álgebras de funciones holomorfas asociadas a sucesiones de ideales de polinomios en abiertos arbitrarios de un espacio de Banach. Como aplicación probamos que el álgebra de funciones holomorfas nucleares de tipo acotado en un conjunto abierto es un álgebra de Fréchet localmente m-convexa. Para el álgebra de funciones de tipo acotado, caracterizamos la envoltura holomorfa en término del espectro. Las evaluaciones en puntos de la envoltura son siempre continuas, pero mostramos un ejemplo de un abierto balanceado de c0 en el que las extensiones a la envoltura no son necesariamente de tipo acotado, respondiendo una pregunta hecha por Hirschowitz. Probamos que para abiertos balanceados y acotados, las extensiones a la envoltura son de tipo acotado. We define the concept of coherent sequence of polynomial ideals on Banach spaces, which allows to relate ideals of homogeneous polynomials of different degrees. To each coherent sequence A, we can associate a FrŽechet space of entire mappings of bounded type, HbA. We extend to HbA a result of Godefroy and Shapiro about hypercyclicity of convolution operators. We also consider the concept of multiplicative sequence of scalar valued polynomial ideals. This allows us to associate an algebra of entire functions of bounded type HbA to each coherent and multiplicative sequence of polynomial ideals A. We prove that, under some natural conditions, the spectrum of the associated algebra, MbA, can be endowed with a structure of Riemann domain over the bidual of the Banach space. Moreover, the extension of each function in HbA to the spectrum is an A-holomorphic function of bounded type in each connected component. We investigate how to define algebras of holomorphic functions associated to sequences of polynomial ideals on arbitrary open sets of a Banach space. As an application we show that the algebra of nuclearly holomorphic functions of bounded type on an open set is a locally m-convex FrŽechet algebra. For the algebra of all bounded type functions, we characterize the envelope of holomorphy in terms of the spectrum of the algebra. The evaluations at points of the envelope are always continuous, but we show an example of a balanced open subset of c0 where the extensions to the envelope are not necessarily of bounded type, answering a question posed by Hirschowitz in 1972. We show that for bounded balanced sets, the extensions to the envelope are of bounded type. Fil: Muro, Santiago. Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales; Argentina.

Given $\u$ a multiplicative sequence of polynomial ideals, we consider the associated algebra of holomorphic functions of bounded type, $H_{b\u}(E)$. We prove that, under very natural conditions verified by many usual classes of polynomials, the spectrum $M_{b\u}(E)$ of this algebra "behaves" like the classical case of $M_{b}(E)$ (the spectrum of $H_b(E)$, the algebra of bounded type holomorphic functions). More precisely, we prove that $M_{b\u}(E)$ can be endowed with a structure of Riemann domain over $E"$ and that the extension of each $f\in H_{b\u}(E)$ to the spectrum is an $\u$-holomorphic function of bounded type in each connected component. We also prove a Banach-Stone type theorem for these algebras., Comment: 19 pages

This paper presents some algorithmic techniques for computing explicitly the Noetherian operators associated with a class of ideals and modules over a polynomial ring. The procedures we include in this work can be easily encoded in computer algebra packages such as CoCoA [CoCoATeam 05].

An element or an ideal of a commutative ring is nilregular if and only if it is regular modulo the nilradical. We prove that if the ring is Noetherian, then every nilregular ideal contains a nilregular element. In constructive mathematics, this proof can then be seen as an algorithm to produce nilregular elements of nilregular ideals whenever the ring is coherent, Noetherian, and discrete. As an application, we give a constructive proof of the Eisenbud--Evans--Storch theorem that every algebraic set in $n$--dimensional affine space is the intersection of $n$ hypersurfaces. The input of the algorithm is an arbitrary finite list of polynomials, which need not arrive in a special form such as a Gr"obner basis. We dispense with prime ideals when defining concepts or carrying out proofs.

A generalised (multivariate) divided difference formula is given for an arbitrary finite set of points with no subsets of three points that lie on a line. This follows from an extension of the Newton’s polynomials and Newton’s interpolation formula. It is derived as the interpolation based on Gr¨obner bases for the grid expressed as a zero-dimensional variety and is typically dependent on the chosen term-ordering and the selected ordering of points in the grid.

We study M(n), the number of distinct values taken by multinomial coefficients with upper entry n, and some closely related sequences. We show that both pP(n)/M(n) and M(n)/p(n) tend to zero as n goes to infinity, where pP(n) is the number of partitions of n into primes and p(n) is the total number of partitions of n. To use methods from commutative algebra, we encode partitions and multinomial coefficients as monomials., Comment: 16 pages, to be published in the Journal of Number Theory

In this paper we present some algorithms for computing an irreducible decomposition of an ideal in a polynomial ring R=K[x1,…,xn] where K is an arbitrary effective field.