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Claude Quitté

Publications and source records attributed to Claude Quitté.

12 recordsLinked to original sources
arXiv2026

Seminormal Rings (following Thierry Coquand)

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 needed to understand the elementary constructive proof for this result given by Thierry Coquand in the paper: On seminormality. J. Algebra 305, no. 1-3, 577-584, (2006). In this new version we have fixed a little typo in Theorem 3.8: the hypothesis seminormal was missing.

math.AC↗
arXiv2025

Heitmann dimension of distributive lattices and commutative rings

This paper is the English translation of the first 4 sections of the article ``Dimension de Heitmann des treillis distributifs et des anneaux commutatifs. Publications Mathématiques de Besançon. Algèbre et théorie des nombres, 2006'', after some corrections. Sections 5-7 of the original article are treated a bit more simply in the book ``Henri Lombardi and Claude Quitté. Commutative algebra: constructive methods. Finite projective modules. Springer, 2015.'' We study the notion of dimension introduced by Heitmann in his remarkable article ``Generating non-Noetherian modules efficiently, Mich. Math. J., 31, (1084)'' as well as a related notion, only implicit in his proofs. We first develop this within the general framework of the theory of distributive lattices and spectral spaces. -- Cet article est une version corrigée des 4 premières sections de l'article ``Dimension de Heitmann des treillis distributifs et des anneaux commutatifs. Publications Mathématiques de Besançon. Algèbre et théorie des nombres, 2006'' Les sections 5 à 7 de l'article original sont traitées de manière un peu plus simple dans ``Henri Lombardi and Claude Quitté. Commutative algebra: constructive methods. Finite projective modules. Springer, 2015.'' Nous étudions la notion de dimension introduite par Heitmann dans son article remarquable ``Generating non-Noetherian modules efficiently, Mich. Math. J., 31, (1084)'', ainsi qu'une notion voisine, seulement implicite dans ses démonstrations. Nous développons ceci d'abord dans le cadre général de la théorie des treillis distributifs et des espaces spectraux. Nous appliquons ensuite cette problématique dans le cadre de l'algèbre commutative.

math.AC↗
arXiv2025

Algebraic identities to prove that a neat finite free algebra is tracically étale

The central objective of this article is to provide an elementary proof of the following theorem, of which we are unaware of any trace in the existing literature. If $B$ is a net finite free algebra over a commutative ring $A$, then it is tracically étale (its trace form is nondegenerate) and a fortiori étale over A. As indicated in the title, our proof is based on algebraic identities. This confirms the implicit adage that much of the most abstract commutative algebra is concentrated in algebraic identities concerning matrices of polynomials over an arbitrary commutative ring. -- -- -- L'objectif central de cet article est de donner une démonstration élémentaire du théorème suivant, dont nous ne connaissons pas de trace dans la littérature existante. Si $B$ est une algèbre libre finie nette sur $A$, alors elle est traciquement étale (sa forme trace est non dégénérée) et à fortiori étale sur $A$. Comme indiqué dans le titre, notre démonstration est basée sur des identités algébriques. Cela confirme l'adage implicite selon lequel une grande partie de l'algèbre commutative la plus abstraite se concentre dans des identités algébriques concernant les matrices de polynômes sur un anneau commutatif arbitraire.

math.AC↗
arXiv2025

Cyclotomic polynomials without using the zeros of $Y^n-1$

This note aims to construct an ``intrinsic'' splitting field for the polynomial $Y^n-1$ over the rational field $\bf Q$, in a way that Gauss, Kummer, Kronecker and Bishop would have liked. Contrary to the usual presentations, our construction does not use any splitting field of $Y^n-1$ which would be given before demonstrating the irreducibility of the cyclotomic polynomial.

math.NT↗
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.

math.AC↗
arXiv2023

Une approche combinatoire pour le résultant multivarié. Le résultant multivarié pour les enfants motivés

We provide, in a 474 pages study, a comprehensive and self-contained treatment of Resultant Theory for a homogeneous system of polynomials with several variables (as many variables as of polynomials). In a non classical way, we use the multiplicative structure of finite free resolutions, by applying it to the complex homogeneous components of the Koszul complex of the system, and this in any degree. Moreover, these complexes have Macaulay decompositions. These three pillars, multiplicative structure, Koszul complex, Macaulay decomposition, allow to establish, surprisingly to us, remarkable binomial relations between 3 families of scalars resulting from the differentials of that complexes. These binomial relations generate several notable results, like a determinantal expression of a certain denominator, depending only on the first differential, and provide in particular formulas expressing the resultant. We have explored more deeply the case of the critical degree delta. This degree produces a fundamental linear form on the homogeneous polynomial component of degree delta, including the resultant. And this fundamental linear form has allowed us to highlight a certain number of remarkable properties. Key words: Elimination theory, homogeneous polynomial system, resultant, determinant of complexes, Cayley determinant, computational algebra, finitefree resolution, Koszul complex, Euler characteristic, Grade, Multiplicative structure, Macaulay decomposition, Fitting invariants, MacRae invariant, regular sequence.

math.AC↗
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].

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.

math.AC↗
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↗
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

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↗