32 citations · 100 across the 24 of their papers we have counts for
24 papers
Seminormal Rings (following Thierry Coquand)
Henri Lombardi, Claude Quitté
The Traverso-Swan theorem says that a reduced ring A is seminormal if and only if the natural morphism from Pic(A) to Pic(A[X]) is an isomorphism. We give here all the details need…
The constructive content of a local-global principle with an application to the structure of a finitely generated projective module
Henri Lombardi
We study the structure of an idempotent matrix over a commutative ring. We make explicit the fundamental system of orthogonal idempotents, hidden in this matrix, for each of wh…
Azumaya algebras and Barr Theorem
Thierry Coquand, Henri Lombardi, Stefan Neuwirth
We study etale topology and the notion of Azumaya algebra over a commutative ring constructively. As an application of the syntactic version of Barr's Theorem, we show the equivale…
The Berlekamp-Massey Algorithm revisited
Nadia Ben Atti, Gema M. Diaz--Toca, Henri Lombardi
We propose a slight modification of the Berlekamp-Massey Algorithm for obtaining the minimal polynomial of a given linearly recurrent sequence. Such a modification enables to expla…
Some remarks about normal rings
Henri Lombardi, Thierry Coquand
We give a constructive proof that is normal when is normal. We apply this result to an operation needed for studying the henselization of a local ring. Our proof is base…
Valuative lattices and spectra
Henri Lombardi, Assia Mahboubi
The first part of the present article consists in a survey about the dynamical constructive method designed using dynamical theories and dynamical algebraic structures. Dynamical m…
Generalized Taylor formulae, computations in real closed valued fields and quantifier elimination
Mari-Emi Alonso, Henri Lombardi
We use generalized Taylor formulae in order to give some simple constructions in the real closure of an \ovfz. We deduce a new, simple quantifier elimination algorithm for \rcvfs a…
Constructive theory of ordinals
Thierry Coquand, Henri Lombardi, Stefan Neuwirth
In Chapter 3 of his Notes on constructive mathematics, Martin-L{ö}f describes recursively constructed ordinals. He gives a constructively acceptable version of Kleene's computable…
Constructive basic theory of central simple algebras
Thierry Coquand, Henri Lombardi, Stefan Neuwirth
We provide a constructive treatment of basic results in the theory of central simple algebras. One main issue is the fact that one starting result, Wedderburn's Theorem stating tha…
On a theorem by de Felipe and Teissier about the comparison of two henselisations in the non-noetherian case
María Emilia Alonso García, Henri Lombardi, Stefan Neuwirth
Let R be a local domain, v a valuation of its quotient field centred in R at its maximal ideal. We investigate the relationship between R^h, the henselisation of R as local ring, a…
Regular entailment relations
Thierry Coquand, Henri Lombardi, Stefan Neuwirth
Inspired by the work of Lorenzen on the theory of preordered groups in the forties and fifties, we define regular entailment relations and show a crucial theorem for this structure…
The syzygy theorem for Bézout rings
Maroua Gamanda, Henri Lombardi, Stefan Neuwirth +1
We provide constructive versions of Hilbert's syzygy theorem for Z and Z/nZ following Schreyer's method. Moreover, we extend these results to arbitrary coherent strict Bézout rings…
Algebra in Bishop's style: some major features of the book "A Course in Constructive Algebra'' by Mines, Richman, and Ruitenburg
Henri Lombardi
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. Clas…
Résolutions libres finies. Méthodes constructives
Thierry Coquand, Henri Lombardi, Claude Quitté +1
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…
Hidden constructions in abstract algebra. Krull Dimension of distributive lattices and commutative rings
Thierry Coquand, Henri Lombardi
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, Españ…
Hidden constructions in abstract algebra, Krull Dimension, Going Up, Going Down
Thierry Coquand, Henri Lombardi
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, Espan…
Virtual Roots of Real Polynomials
Laureano Gonzalez--Vega, Henri Lombardi, Louis Mahé
The fact that a real univariate polynomial misses some real roots is usually overcame by considering complex roots, but the price to pay for, is a complete lost of the sign structu…
Roger Apéry: humour at the service of a free and original thinking on constructive mathematics
Henri Lombardi, Stefan Neuwirth
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 e…
Dynamical method in algebra: Effective Nullstellensätze
Michel Coste, Henri Lombardi, Marie-Françoise Roy
We give a general method for producing various effective Null and Positivstellensätze, and getting new Positivstellensätze in algebraically closed valued fields and ordered groups.…
Lattice-ordered groups generated by an ordered group and regular systems of ideals
Thierry Coquand, Henri Lombardi, Stefan Neuwirth
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 Lo…
Algèbre commutative Méthodes constructives
Henri Lombardi, Claude Quitté
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…
Lipschitz continuity in the Hurst parameter of functionals of stochastic differential equations driven by a fractional Brownian motion
Alexandre Richard, Denis Talay
Sensitivity analysis w.r.t. the long-range/memory noise parameter for probability distributions of functionals of solutions to stochastic differential equations is an important sto…
Local Bézout Theorem for Henselian rings
M. -Emilia Alonso, Henri Lombardi
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 hypot…
An elementary recursive bound for effective Positivstellensatz and Hilbert 17-th problem
Henri Lombardi, Daniel Perrucci, Marie-Françoise Roy
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 rat…