arXiv subjects

Henri Lombardi

Publications and source records attributed to Henri Lombardi.

49 records · Page 3Linked to original sources

arXiv2019

Algèbre commutative Méthodes constructives

This book is an introductory course to basic commutative algebra with a particular emphasis on finitely generated projective modules, which constitutes the algebraic version of the vector bundles in differential geometry. We adopt the constructive point of view, with which all existence theorems have an explicit algorithmic content. In particular, when a theorem affirms the existence of an object -- the solution of a problem -- a construction algorithm of the object can always be extracted from the given proof. We revisit with a new and often simplifying eye several abstract classical theories. In particular, we review theories which did not have any algorithmic content in their general natural framework, such as Galois theory, the Dedekind rings, the finitely generated projective modules or the Krull dimension. Constructive algebra is actually an old discipline, developed among others by Gauss and Kronecker. We are in line with the modern "bible" on the subject, which is the book by Ray Mines, Fred Richman and Wim Ruitenburg, A Course in Constructive Algebra, published in 1988.

math.AC↗

arXiv2019

Algebra in Bishop's style: some major features of the book "A Course in Constructive Algebra'' by Mines, Richman, and Ruitenburg

The book "A Course in Constructive Algebra" (1988) shows the way of understanding classical basic algebra in a constructive style similar to Bishop's Constructive Mathematics. Classical theorems are revisited, with a new flavour, and become much more precise. We are often surprised to find proofs that are simpler and more elegant than the usual ones. In fact, when one cannot use magic tools as the law of excluded middle, it is necessary to understand what is the true content of a classical proof. Also, usual shortcuts allowed in classical proofs introduce sometimes useless detours. In order to understand clearly a problem, prescience may be a handicap.

math.HO↗

arXiv2018

Résolutions libres finies. Méthodes constructives

In this memoir, we give a completely constructive version of the celebrate book 'Finite Free Resolutions' by Northcott, and of some other results related to the depth à la Hochster, the Cayley complexes and their determinants, and the finite projective resolutions.

math.AC↗

arXiv2018

Lattice-ordered groups generated by an ordered group and regular systems of ideals

Unbounded entailment relations, introduced by Paul Lorenzen (1951), are a slight variant of a notion which plays a fundamental rôle in logic (see Scott 1974) and in algebra (see Lombardi and Quitté 2015). We call systems of ideals their single-conclusion counterpart. If they preserve the order of a commutative ordered monoid G and are equivariant w.r.t. its law, we call them equivariant systems of ideals for G: they describe all morphisms from G to meet-semilattice-ordered monoids generated by (the image of) G. Taking an article by Lorenzen (1953) as a starting point, we also describe all morphisms from a commutative ordered group G to lattice-ordered groups generated by G through unbounded entailment relations that preserve its order, are equivariant, and satisfy a regularity property invented by Lorenzen (1950); we call them regular entailment relations. In particular, the free lattice-ordered group generated by G is described through the finest regular entailment relation for G, and we provide an explicit description for it; it is order-reflecting if and only if the morphism is injective, so that the Lorenzen-Clifford-Dieudonné theorem fits into our framework. Lorenzen's research in algebra starts as an inquiry into the system of Dedekind ideals for the divisibility group of an integral domain R, and specifically into Wolfgang Krull's "Fundamentalsatz" that R may be represented as an intersection of valuation rings if and only if R is integrally closed: his constructive substitute for this representation is the regularisation of the system of Dedekind ideals, i.e. the lattice-ordered group generated by it when one proceeds as if its elements are comparable.

math.LO↗

arXiv2018

Modules projectifs de type fini, applications linéaires croisées et inverses généralisés

We give a general theory of generalised inverses and we explain the link with the theory of finitely generated projective modules. All the paper is written in constrctive mathematics in Bishop style. So all results do have a clear algorithmic content. We give also a complexity analysis of the algorihms corresponding to the main theorems. Here is a more detailed abstract in french: D'une part, nous développons la théorie générale des inverses généralisés de matrices en la mettant en rapport avec la théorie constructive des modules projectifs de type fini. D'autre part nous précisons certains aspects de cette théorie liés au calcul formel et à l'analyse numérique matricielle. Nous démontrons en particulier qu'on peut tester si un $\bf A$-module de présentation finie est projectif et calculer une matrice de projection correspondante "en temps polynomial". Plus précisément pour une matrice $A\in {\bf A}^{m \times n}$ on peut décider s'il existe un inverse généralisé $B$ pour $A$ (i.e. une matrice $B$ vérifiant $ABA=A$ et $BAB=B$) et, en cas de réponse positive, calculer un tel inverse généralisé par un algorithme qui utilise $O(p^6\,q^{2})$ opérations arithmétiques (avec $p=\inf(m,n)$, $q=\sup(m,n)$) et un nombre polynomial de tests d'appartenance d'un élément à un idéal engendré par "un petit nombre d'éléments".

math.AC↗

arXiv2018

Constructions cachées en algèbre abstraite. Dimension de Krull, Going Up, Going Down (revised version, 2008)

Nous rappelons des versions constructives de la théorie de la dimension de Krull dans les anneaux commutatifs et dans les treillis distributifs, dont les bases ont été posées par Joyal, Espanõl et les deux auteurs. Nous montrons sur les exemples de la dimension des algèbres de présentation finie, du Going Up, du Going Down \ldots que cela nous permet de donner une version constructive de grands théorèmes classiques, et par conséquent de récupérer un contenu calculatoire explicite lorsque ces théorèmes abstraits sont utilisés pour démontrer l'existence d'objets concrets. Nous pensons ainsi mettre en oeuvre une réalisation partielle du programme de Hilbert pour l'algèbre abstraite classique. We present constructive versions of Krull's dimension theory for commutative rings and distributive lattices. The foundations of these constructive versions are due to Joyal, Espanõl and the authors. We show that this gives a constructive version of basic classical theorems (dimension of finitely presented algebras, Going up and Going down theorem, \ldots), and hence that we get an explicit computational content when these abstract results are used to show the existence of concrete elements. This can be seen as a partial realisation of Hilbert's program for classical abstract commutative algebra.

math.AC↗

arXiv2017

Hidden constructions in abstract algebra. Krull Dimension of distributive lattices and commutative rings

We present constructive versions of Krull's dimension theory for commutative rings and distributive lattices. The foundations of these constructive versions are due to Joyal, Español and the authors. We show that the notion of Krull dimension has an explicit computational content in the form of existence (or lack of existence) of some algebraic identities. We can then get an explicit computational content where abstract results about dimensions are used to show the existence of concrete elements. This can be seen as a partial realisation of Hilbert's program for classical abstract commutative algebra.

math.AC↗

arXiv2017

Hidden constructions in abstract algebra, Krull Dimension, Going Up, Going Down

We present constructive versions of Krull's dimension theory for commutative rings and distributive lattices. The foundations of these constructive versions are due to Joyal, Espanõl and the authors. We show that this gives a constructive version of basic classical theorems (dimension of finitely presented algebras, Going up and Going down theorem, \ldots), and hence that we get an explicit computational content where these abstract results are used to show the existence of concrete elements. This can be seen as a partial realisation of Hilbert's program for classical abstract commutative algebra.

math.AC↗

arXiv2017

Roger Apéry: humour at the service of a free and original thinking on constructive mathematics

This article is an invitation to read a famous text of Roger Ap{é}ry, "Math{é}matique constructive", published in the book "Penser les math{é}matiques: s{é}minaire de philosophie et math{é}matiques de l'{É}cole normale sup{é}rieure (J. Dieudonn{é}, M. Loi, R. Thom)" [arXiv:1705.05581]. It also proposes a commented bibliography for constructive mathematics and a technical note on measure theory.

math.HO↗

arXiv2017

An elementary recursive bound for effective Positivstellensatz and Hilbert 17-th problem

We prove elementary recursive bounds in the degrees for Positivstellensatz and Hilbert 17-th problem, which is the expression of a nonnegative polynomial as a sum of squares of rational functions. We obtain a tower of five exponentials. A precise bound in terms of the number and degree of the polynomials and their number of variables is provided in the paper.

math.AG↗

arXiv2016

Local Bézout Theorem for Henselian rings

This paper gives an elementary proof of an improved version of the algebraic Local Bézout Theorem (given by the authors in JSC 45 (2010) 975--985). Here we remove some ad hoc hypotheses and obtain an optimal algebraic version of the theorem. Given a system of $n$ polynomials in $n$ indeterminates with coefficients in a local normal domain $(A, m,k)$ with an algebraically closed quotient field, which residually defines an isolated point in $k^n$ of multiplicity $r$, we prove that there are finitely many zeroes of the system above the residual zero (i.e., with coordinates in $m$), and the sum of their multiplicities is $r$. Our proof is based on the border basis technique of computational algebra. Here we state and prove an {\em algebraic version} of this theorem in the setting of arbitrary Henselian rings and $m$-adic topology. We are somehow inspired by Arnold, exploiting an abstract version of Weierstrass division (in a Henselian ring) and we introduce also an abstract version of which he called the "multilocal ring". Roughly speaking we consider a finitely presented $A$-algebra, where $(A, m, k)$ is a local ring such that the special point is a $k$-algebra with an isolated zero of multiplicity $r$ and we prove that the "multilocal ring" determined by this point is a free $A$-module of rank $r$.

math.AC↗

arXiv2016

Méthodes Matricielles Introduction à la Complexité Algébrique

The main purpose of this book is to propose an introduction to the modern tools of algebraic complexity. To remain as simple as possible while providing meaningful examples, we chose to focus on effective linear algebra; this is certainly one of the best possible choices to give an idea of the main problems in algebraic complexity. The contents of the book are the following: classical results of linear algebra and basic algorithms in linear algebra, straight-line programs as a model of computation, with an emphasis on Strassen's method known as "elimination of divisions", a discussion on various notions of complexity, a presentation of the general algorithmic strategy "divide and conquer", a first important example: the fast multiplication of polynomials, the very heart of the book: the fast multiplication of matrices, with a discussion of derived fast algorithms for various problems of linear algebra. The particular case of the computation of the determinant is then discussed, and the last chapter deals with the difficult computation of the permanent (Valiant's conjecture). This book gives an introduction to a fascinating subject at a reasonable level. . . for French-speaking people.

math.AC↗

arXiv2015

Calcul matriciel généralisé sur les domaines de Prüfer

In this paper, we first present an algorithm for computing the Hermite normal form of pseudo-matrices over Prüfer domains. This algorithm allows us to provide constructive proofs of the main theoretical results on finitely presented modules over Prüfer domains and to discuss the resolution of linear systems. In some sense, we generalize the methodology developed by Henri Cohen for Dedekind domains. Finally, we present some results over Prüfer domains of dimension one about the Smith normal form.

math.AC↗