Computational Methods for the Construction of a Class of Noetherian Operators

This paper presents some algorithmic techniques to compute explicitly the noetherian operators associated to 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 and Singular

La queste del saint Gra(AL): A computational approach to local algebra

AbstractWe show how, by means of the Tangent Cone Algorithm, the basic functions related to the maximal ideal topology of a local ring can be effectively computed in the situations of geometrical significance, i.e.:(1)localizations of coordinate rings of a variety at the prime ideal defining a subvariety,(2)rings of algebraic formal power series rings.In particular we show how the method of “associated graded rings” can be turned into an effective tool to compute local algebraic invariants of varieties

Computing Limit Points of Quasi-components of Regular Chains and its Applications

Computing limit is a fundamental task in mathematics and different mathematical concepts are defined in terms of limit computations. Among these mathematical concepts, we are interested in three different types of limit computations: first, computing the limit points of solutions of polynomial systems represented by regular chains, second, computing tangent cones of space curves at their singular points which can be viewed as computing limit of secant lines, and third, computing the limit of real multivariate rational functions. For computing the limit of solutions of polynomial systems represented by regular chains, we present two different methods based on Puiseux series expansions and linear changes of coordinates. The first method, which is based on Puiseux series expansions, addresses the problem of computing real and complex limit points corresponding to regular chains of dimension one. The second method studies regular chains under changes of coordinates. It especially computes the limit points corresponding to regular chains of dimension higher than one for some cases. we consider strategies where these changes of coordinates can be either generic or guided by the input. For computing the Puiseux parametrizations corresponding to regular chains of dimension one, we rely on extended Hensel construction (EHC). The Extended Hensel Construction is a procedure which, for an input bivariate polynomial with complex coefficients, can serve the same purpose as the Newton-Puiseux algorithm, and, for the multivariate case, can be seen as an effective variant of Jung-Abhyankar Theorem. We show that the EHC requires only linear algebra and univariate polynomial arithmetic. We deduce complexity estimates and report on a software implementation together with experimental results. We also outline a method for computing the tangent cone of a space curve at any of its points. We rely on the theory of regular chains and Puiseux series expansions. Our approach is novel in that it explicitly constructs the tangent cone at arbitrary and possibly irrational points without using a Standard basis. We also present an algorithm for determining the existence of the limit of a real multivariate rational function q at a given point which is an isolated zero of the denominator of q. When the limit exists, the algorithm computes it, without making any assumption on the number of variables. A process, which extends the work of Cadavid, Molina and V´elez, reduces the multivariate setting to computing limits of bivariate rational functions. By using regular chain theory and triangular decomposition of semi-algebraic systems, we avoid the computation of singular loci and the decomposition of algebraic sets into irreducible components

Géométrie des espaces de tenseursUne approche effective appliquée à la mécanique des milieux continus

Tensorial formulation of mechanical constitutive equations is a very important matterin continuum mechanics. For instance, the space of elastic tensors is a subspace of 4thorder tensors with a natural SO(3) group action. More generaly, we have to study thegeometry of a tensor space defined on R 3 , under O(3) group action.To describe such a geometry, we first have to exhibit its isotropy classes, also namedsymetry classes. Indeed, each tensor space possesses a finite number of isotropy classes.In this present work, we propose an original method to obtain isotropy classes of a giventensor space. As an illustration of this new method, we get for the first time the isotropyclasses of a 8th order tensor space occuring in second strain-gradient elasticity theory.In the case of a real representation of a compact group, invariant algebra seperatesthe orbits. This observation motivates the purpose to find a finite generating set of poly-nomial invariants. For that purpose, we make use of the link between tensor spaces andspaces of binary forms, which belongs to the classical invariant theory. We thus have todeal with SL(2, C) group action. To obtain new results, we have reformulated and rein-terpreted effective approaches of Gordan’s algorithm, developped during XIXth century.We then obtain for the first time a minimal generating family of elasticity tensor space,and a generating family of piezoelectricity tensor space. Using linear algebra arguments,we were also able to get important relations of classical invariant theory, such as theGordan’s series and the Abdesselam–Chipalkatti’s quadratic relations on transvectants.Plusieurs lois de comportement mécaniques possèdent une formulation tensorielle,comme c’est par exemple le cas pour l’étude des matériaux élastiques. Dans ce cas in-tervient un sous-espace de tenseurs d’ordre 4, noté Ela et appelé espace des tenseursd’élasticité. Les questions de classification des matériaux élastiques passent alors par lanécessité de décrire les orbites de l’espace Ela sous l’action du groupe SO(3). Plus gé-néralement, on est amené à étudier la géométrie d’un espace de tenseurs sur R 3 , vial’action du groupe O(3).Cette géométrie est tout d’abord caractérisée par ses différentes classes d’isotropies,encore appelées classes de symétries. Chaque espace de tenseurs possède en effet unnombre fini de classes d’isotropies. Nous proposons dans notre travail une méthode ori-ginale et générale pour obtenir les classes d’istropie d’un espace de tenseurs quelconque.Nous avons ainsi pu obtenir pour la première fois les classes d’isotropie d’un espace detenseurs d’ordre 8 intervenant en théorie de l’élasticité linéaire du second-gradient de ladéformation.Dans le cas d’une représentation réelle d’un groupe compact, l’algèbre des polynômesinvariants sépare les orbites, ce qui motive donc la recherche d’une famille génératriceminimale de polynômes invariants. Celle-ci se fait en exploitant le lien existant entreles espaces de tenseurs et les espaces de formes binaires et plus précisément la théorieclassique des invariants. On ne fait donc plus intervenir le groupe SO(3) mais le groupeSL(2, C). Nous avons ainsi repris et ré-interprété les approches effectives de cette théo-rie, notamment développées par Gordan au XIX e siècle. Cette ré-interprétation nous apermis d’obtenir de nombreux résultats, notamment la détermination d’une famille gé-nératrice minimale d’invariants pour l’élasticité mais aussi pour la piézoélectricté. No-tons aussi que nous avons pu retrouver d’une façon simple des relations importantesintervenant en théorie classique des invariants, à savoir les fameuses séries de Gordan,ainsi que des relations plus récentes d’Abdesselam–Chipalkatti sur les transvectants deformes binaires