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Henri Lombardi

Publications and source records attributed to Henri Lombardi.

At least 37 records · Page 2Linked to original sources
arXiv2024

Geometric theories for real number algebra without sign test or dependent choice axiom

In this memoir, we seek to construct a constructive theory that is as complete as possible to describe the algebraic properties of the real number field in constructive mathematics without a dependent choice axiom. To this purpose, we use a dynamical version of geometric theories. We obtain a nice description of the algebraic properties of the real number field, but also a first outline for a constructive theory of certain o-minimal structures. The memoir we present here is an unfinished development of the article by the authors https://inria.hal.science/hal-01426164. Compared to that paper, however, we have modified the definition of continuous semialgebraic functions, in the same spirit in which Bishop defines a continuous real function as a uniformly continuous function on any bounded interval. Despite its unfinished nature and the many questions that we do not currently know how to answer, we hope that this paper will arouse interest for its original approach to the subject. This paper is an English translation of a French version on arXiv:2406.15218

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arXiv2024

Commutative algebra: Constructive methods. Finite projective modules

This book is an introductory course to basic commutative algebra with a particular emphasis on finitely generated projective modules. We adopt the constructive point of view, with which all existence theorems have an explicit algorithmic content 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 domains, the finitely generated projective modules or the Krull dimension.

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arXiv2024

The syzygy theorem for Bézout rings

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 with a divisibility test for the case of finitely generated modules whose module of leading terms is finitely generated.

math.AC↗
arXiv2024

Anneaux à diviseurs et anneaux de Krull (une approche constructive)

We give give an elementary and constructive version of the theory of "Prüfer v-Multiplication Domains" (which we call "anneaux à diviseurs" in the paper) and Krull Domains. The main results of these theories are revisited from a constructive point of view, following the Bishop style, and without assuming properties of complete factorizations. Nous présentons dans cet article une approche constructive, dans le style de Bishop, de la théorie des diviseurs et des anneaux de Krull. Nous accordons une place centrale aux "anneaux à diviseurs," appelés PvMD dans la littérature anglaise. Les résultats classiques sont obtenus comme résultats d'algorithmes explicites sans faire appel aux hypothèses de factorisation complète.

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arXiv2023

Valuative lattices and spectra

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 methods uncovers a hidden computational content for numerous abstract objects of classical mathematics, which seem a priori inaccessible constructively, e.g., the algebraic closure of a (discrete) field. When a proof in classical mathematics uses these abstract objects and results in a concrete outcome, dynamical methods generally make possible to discover an algorithm for this concrete outcome. The second part of the article applies this dynamical method to the theory of divisibility. We compare two notions of valuative spectra present in the literature and we introduce a third notion, which is implicit in an article devoted to the dynamical theory of algebraically closed discrete valued fields. The two first notions are respectively due to Huber \& Knebusch and to Coquand. We prove that the corresponding valuative lattices are essentially the same. We establish formal Valuativestellensätze corresponding to these theories, and we compare the various resulting notions of valuative dimensions.

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arXiv2023

Dimension de Krull, Nullstellensätze et Évaluation Dynamique

We prove constructively a Nullstellensatz giving an equivalence between the existence of a certain kind of algebraic identity on one hand, and the impossibility of finding an increasing sequence of irreducible varieties obeying certain constraints on the other hand. The ususal Nullstellensatz corresponds to the case of varieties that are reduced to a point. We settle also a similar formal Nullstellensatz related to increasing sequences of primes. An important particular case is given by the notion of pseudo regular sequence. This allows a new characterisation of the Krull dimension of a ring. This characterisation via pseudo regular sequences is elementary and constructive. Our method uses dynamical algebraic structures which were introduced in previous papers.

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arXiv2023

The constructive content of a local-global principle with an application to the structure of a finitely generated projective module

We study the structure of an idempotent matrix $F$ over a commutative ring. We make explicit the fundamental system of orthogonal idempotents, hidden in this matrix, for each of which the matrix has a well-defined rank. Similarly we find a finite number of comaximal elements of the ring which make explicit the fact that the codomain of $F$ is locally free. Our proofs are based on the abstract local-global principle. We give two methods to recover a constructive proof of these results. The most interesting one is a constructive interpretation of a very simple version of the abstract local-global principle. We think we have made a significant step towards a constructive version of the "Hilbert programme" for abstract algebra, i.e. the automatic translation of proofs of abstract algebra into constructive proofs.

math.AC↗
arXiv2023

Constructive basic theory of central simple algebras

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 that a simple algebra is a matrix algebra over a skew field, is not constructively valid. We solve this problem by proving instead a dynamical version of this theorem. One can use this to give constructive proofs of basic results of the theory of central simple algebras, such as Skolem-Noether Theorem. We illustrate this development by giving an elementary constructive proof of a theorem of Becher (which is itself a consequence of a celebrated theorem of Merkurjev).

math.RA↗
arXiv2023

An algorithm for computing syzygies on $V[X]$ when $V$ is a valuation domain

We give an algorithm for computing the V-saturation of any finitely-generated submodule of a power of V[X], where V is a valuation domain. Our algorithm is based on a notion of "echelon form" which ensures its correctness. This allows us to compute a finite system of generators for the syzygy module of any finitely generated submodule of a power of V[X].

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arXiv2022

Some remarks about normal rings

We give a constructive proof that $R[X]$ is normal when $R$ is normal. We apply this result to an operation needed for studying the henselization of a local ring. Our proof is based on the case where $R$ is without zero divisors, which is more involved than the case where $R$ is an integral domain. We have to use a constructive deciphering technique that replaces the use of minimal primes (in classical mathematics) by suitable explicit localizations in a suitable tree.

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arXiv2022

Dynamic computations inside the algebraic closure of a valued field

We explain how to compute in the algebraic closure of a valued field. These computations heavily rely on the \NPAz. They are made in the same spirit as the dynamic algebraic closure of a field. They give a concrete content to the theorem saying that a valued field does have an algebraically closed valued extension. The algorithms created for that purpose can be used to perform an effective quantifier elimination for algebraically closed valued fields, which relies on a very natural geometric idea.

math.AC↗
arXiv2022

Construction du hensélisé d'un corps valué

We give an explicit construction of the henselization of a valued field, with a constructive proof. It is analogous to the construction of the real closure of a discrete ordered field. Nous donnons une construction explicite, et constructivement prouvée, du hensélisé d'un corps valué. Cette construction peut être considérée comme l'analogue, dans le cas valué, de la construction de la clôture rééelle d'un corps ordonné.

math.AC↗
arXiv2022

Dimension de Heitmann des treillis distributifs et des anneaux commutatifs

We study a notion of dimension which was introduced by R. Heitmann in his remarkable paper in 1984, and also a related notion, implicit in the proofs in his paper. We develop these notions in the general framework of distributive lattices and spectral spaces. We obtain in this way constructive versions of important theorems in commutative algebra, with simpler proofs than the classical ones, and some new results.

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arXiv2021

Virtual Roots of Real Polynomials

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 structure that a set of real roots is endowed with (mutual position on the line, signs of the derivatives, etc...). In this paper we present real substitutes for these missing roots which keep sign properties and which extend of course the existing roots. Moreover these "virtual roots" are the values of semialgebraic continuous -- rather uniformly -- functions defined on the set of monic polynomials. We present some applications.

math.AG↗
arXiv2021

On a theorem by de Felipe and Teissier about the comparison of two henselisations in the non-noetherian case

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, and {v}, the henselisation of the valuation v, by focussing on the recent result by de Felipe and Teissier referred to in the title. We give a new proof that simplifies the original one by using purely algebraic arguments. This proof is moreover constructive in the sense of Bishop and previous work of the authors, and allows us to obtain as a by-product a (slight) generalisation of the theorem by de Felipe and Teissier.

math.AC↗
arXiv2020

Regular entailment relations

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. We also describe equivariant systems of ideals {à} la Lorenzen and show that the remarkable regularisation process invented by him yields a regular entailment relation. By providing constructive objects and arguments, we pursue Lorenzen's aim of "bringing to light the basic, pure concepts in their simple and transparent clarity"

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