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Cluster algebras are integral domains with a particular combinatorial structure. This structure consists in the data of a family of seeds linked together by an operation called mutation. Each seed consists in two parts : acluster and a quiver.Richardson open varieties are some strata of the flag variety associated to a simple linear algebraic group of simply-laced type. These are the intersection of Schubert cells with respect to two opposite Borel subgroups.In [Lec16] a cluster subalgebra of maximal rank on the coordinate ring of an open Richardson variety has been constructed and this subalgebra is conjectured to be equal to the whole ring. The construction of this clusteralgebra comes from a Frobenius category C v,w of modules over the preprojective algebra, defined as the intersection of two categories C w and C v already studied by Geiss, Leclerc, Schröer and Buan, Iyama, Reiten and Scott. The bond between cluster algebras and cluster structures is given by the cluster character defined in [GLS06]. In this thesis we build an algorithm which, given the parameters defining a Richardson open variety, compute an explicit maximal rigid module of the associated Frobenius category and its quiver. This algorithm has an initial seed for the cluster structure on C w defined by a representative w of an element w of the Weyl group as a starting datum. By a combinatorially defined sequence of mutation on this initial seed we obtain a maximal rigid module of C w which is, up to deletion of some direct summands is a maximal rigid module of C v,w . In addition, the subquiver of the mutated quiver is exactly the quiver of the endomorphism algebra of the C v,w -maximal rigid module, giving then the complete description of an initial seed for the cluster structure on C v,w ., Les algèbres amassées sont des anneaux commutatifs intègres avec une structure combinatoire particulière.Cette structure consiste en la donnée d’une famille de graines, liées entre elles par une opération appelée mutation.Chaque graine est composée de deux parties : un amas et un carquois.Les variétés de Richardson ouvertes sont des strates de la variété de drapeaux associée à un groupe linéaire algébrique de type simplement lacé. Elles sont l’intersection de cellules de Schubert respectivement à deux sous- groupes de Borel opposés. Dans [Lec16], une sous-algèbre amassée de rang maximal sur l’anneau de coordonnées d’une variété de Richardson ouverte a été construite et cette sous-algèbre est conjecturée être égale à l’anneau entier. La construction de cette algèbre amassée provient d’une catégorie de Frobenius C v,w de modules sur l’algèbre préprojective, définie comme intersection de deux catégories C w et C v déjà étudiées par Geiss, Leclerc, Schröer et Buan, Iyama, Reiten et Scott. Le lien entre les algèbres amassées et les structures amassées est donné par le caractère d’amas défini dans [GLS06].Dans cette thèse, nous construisons un algorithme qui, étant donné les paramètres définissant une variété de Richardson ouverte, construit un module rigide maximal explicite de la catégorie de Frobenius associée et soncarquois. Cet algorithme a pour donnée de départ la graine initiale pour la structure amassée sur C w définie par un représentant w d’un élément w du groupe de Weyl. Par le biais d’une suite de mutations déterminéecombinatoirement, on obtient à partir de la graine initiale un module rigide maximal de C w qui, à suppression de certains facteurs directs près, est un module rigide maximal de C v,w . De plus le sous-carquois du carquois muté est exactement le carquois de l’algèbre d’endomorphisme du module rigide maximal de C v,w donnant alors la description complète d’une graine initiale pour la structure amassée de C v,w .

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.

The thesis consists of 6 papers. (1) We calculate the generators of SA₃(ℝ)-invariants for parabolic surfaces. (2) We calculate rigid relative invariants for rigid constant Levi-rank 1 and 2-non-degenerate hypersurfaces in ℂ³: V₀, I₀, Q₀ having 11, 52, 824 monomials in their numerators. (3) We organize all affinely homogeneous nondegenerate surfaces in ℂ³ in inequivalent branches. (4) For a directed harmonic current near a non-hyperbolic linearized singularity which does not give mass to any of the trivial separatrices and whose trivial extension across 0 is ddc-closed, we show that the Lelong number at 0 is: 4.1) strictly positive if the eigenvalue λ>0; 4.2) zero if λ is a negative rational number; 4.3) zero if λ=2. (6) We show that circulant matrices having k ones and k+1 zeros in the first row are always nonsingular when 2k+1 is either a power of a prime, or a product of two distinct primes. For any other integer 2k+1 we exhibit a singular circulant matrix.; La thèse se compose de 6 articles. (1) Nous calculons les générateurs des SA₃(ℝ)-invariants pour les surfaces paraboliques. (2) Nous calculons les invariants rigides relatifs pour les hypersurfaces rigides 2-non-dégénérées de rang de Levi constant 1 dans ℂ³: V₀, I₀, Q₀ ayant 11, 52, 824 monômes au numérateur. (3) Nous organisons tous les modèles affinement homogènes non-dégénérés dans ℂ³ en branches inéquivalentes. (4) Pour un courant harmonique dirigé autour d'une singularité linéarisée non-hyperbolique qui ne charge pas les séparatrices triviales dont l'extension triviale à travers 0 est ddc-fermée, nous démontrons que le nombre de Lelong en 0 est : 4.1) strictement positif si λ>0 ; 4.2) nul si λ est rationnel et négatif ; 4.3) nul si λ est négatif et si T est invariant sous l'action d'un sous-groupe cofini du groupe de monodromie. (5) Nous construisons des fibrés holomorphes en droites en toute dimension n>=2 non-prolongeables au sens de Hartogs. (6) Nous montrons que les matrices circulantes ayant k entrées 1 et k+1 entrées 0 dans leur première rangée sont toujours non singulières lorsque 2k+1 est soit une puissance d'un nombre premier, soit un produit de deux nombres premiers distincts. Pour tout autre entier 2k+1, nous exhibons une matrice circulante singulière.

Using Galois theory of functional equations, we give a new proof of the main result of the paper "Transcendental transcendency of certain functions of Poincar\'e" by J.F. Ritt, on the differential transcendence of the solutions of the functional equation R(y(t))=y(qt), where R is a rational function with complex coefficients which verifies R(0)=0, R'(0)=q, where q is a complex number with |q|>1. We also give a partial result in the case of an algebraic function R.

We generalize the notion of Davenport constants to a `higher degree' and obtain various lower and upper bounds, which are sometimes exact as is the case for certain finite commutative rings of prime power cardinality. Two simple examples that capture the essence of these higher degree Davenport constants are the following. 1) Suppose $n = 2^k$, then every sequence of integers $S$ of length $2n$ contains a subsequence $S'$ of length at least two such that $\sum_{a_i,a_j \in S'} a_ia_j \equiv 0 \pmod{n}$ and the bound is sharp. 2) Suppose $n \equiv1 \pmod{2}$, then every sequence of integers $S$ of length $2n -1$ contains a subsequence $S'$ of length at least two such that $\sum_{a_i,a_j \in S'} a_ia_j \equiv 0 \pmod{n}$. These examples illustrate that if a sequence of elements from a finite commutative ring is long enough, certain symmetric expressions have to vanish on the elements of a subsequence., Comment: 16 pages

We prove that the resultant of two “sufficiently generic” bivariate polynomials over a finite field can be computed in quasi-linear expected time, using a randomized algorithm of Las Vegas type. A similar complexity bound is proved for the computation of the lexicographical Grobner basis for the ideal generated by the two polynomials.

International audience; We discuss the application of random projections to conic programming: notably linear, second-order and semidefinite programs. We prove general approximation results on feasibility and optimality using the framework of formally real Jordan algebras. We then discuss some computational experiments on randomly generated semidefinite programs in order to illustrate the practical applicability of our ideas.

The goal is to show how to obtain Molien functions for SO(3) invariant and covariant modules, how to recast them in an appropriate form when these modules are not free, and how to use them to build integrity bases for free modulesor their generalization in the case of non-free covariants modules. We also explain how to easily derive O(3) invariants and covariants basis from SO(3) ones.However, applications of SO(3) invariant and covariant bases extends to cases such as the modelling of potential energy or dipole moment hypersurfaces in quantum chemistry,where O(3)-symmetry is expected to hold, unless parity violation is considered, but where the use of the SO(3) invariant ring is more practical than that of O(3).

In this paper, we study the algebraic relations satisfied by the solutions of q-difference equations and their transforms with respect to an auxiliary operator. Our main tools are the parametrized Galois theories developed in Hardouin and Singer (Math Ann 342(2):333–377, 2008) and Ovchinnikov and Wibmer (Int Math Res Not 12:3962–4018, 2015). The first part of this paper is concerned with the case where the auxiliary operator is a derivation, whereas the second part deals with a $$\mathbf {q}$$ -difference operator. In both cases, we give criteria to guarantee the algebraic independence of a series, solution of a q-difference equation, with either its successive derivatives or its $$\mathbf {q}$$ -transforms. We apply our results to q-hypergeometric series.

A complete theory of the degree of a polynomial ideal is presented, with a systematic use of the rational form of the Hilbert function in place of the (more commonly used) Hilbert polynomial. This is used for a simple algebraic proof of classical Bézout theorem, and for proving a "strong Bézout inequality", which has as corollaries all previously known Bézout inequalities , and is much sharper than all of them in the case of a non-equidimensional ideal.

Let $F\in \mathbb{K}[X, Y ]$ be a polynomial of total degree $D$ defined over a perfect field $\mathbb{K}$ of characteristic zero or greater than $D$. Assuming $F$ separable with respect to $Y$ , we provide an algorithm that computes the singular parts of all Puiseux series of $F$ above $X = 0$ in less than $\tilde{\mathcal{O}}(D\delta)$ operations in $\mathbb{K}$, where $\delta$ is the valuation of the resultant of $F$ and its partial derivative with respect to $Y$. To this aim, we use a divide and conquer strategy and replace univariate factorization by dynamic evaluation. As a first main corollary, we compute the irreducible factors of $F$ in $\mathbb{K}[[X]][Y ]$ up to an arbitrary precision $X^N$ with $\tilde{\mathcal{O}}(D(\delta + N ))$ arithmetic operations. As a second main corollary, we compute the genus of the plane curve defined by $F$ with $\tilde{\mathcal{O}}(D^3)$ arithmetic operations and, if $\mathbb{K} = \mathbb{Q}$, with $\tilde{\mathcal{O}}((h+1)D^3)$ bit operations using a probabilistic algorithm, where $h$ is the logarithmic heigth of $F$., Comment: 27 pages, 2 figures

We consider pairs of automorphisms $(\phi,\sigma)$ acting on fields of Laurent or Puiseux series: pairs of shift operators $(\phi\colon x\mapsto x+h_1, \sigma\colon x\mapsto x+h_2)$, of $q$-difference operators $(\phi\colon x\mapsto q_1x,\ \sigma\colon x\mapsto q_2x)$, and of Mahler operators $(\phi\colon x\mapsto x^{p_1},\ \sigma\colon x\mapsto x^{p_2})$. Given a solution $f$ to a linear $\phi$-equation and a solution $g$ to a linear $\sigma$-equation, both transcendental, we show that $f$ and $g$ are algebraically independent over the field of rational functions, assuming that the corresponding parameters are sufficiently independent. As a consequence, we settle a conjecture about Mahler functions put forward by Loxton and van der Poorten in 1987. We also give an application to the algebraic independence of $q$-hypergeometric functions. Our approach provides a general strategy to study this kind of question and is based on a suitable Galois theory: the $\sigma$-Galois theory of linear $\phi$-equations.

International audience; Krull's Fundamentalsatz, the generalisation of the main theorem of elementary number theory to integral domains, is the starting point of Lorenzen’s career in mathematics. This article traces a conceptual history of Lorenzen's successive reformulations of the Fundamentalsatz on the basis of excerpts of his articles. An edition of the extant correspondence of Lorenzen with Hasse, Krull, and Aubert provides a better understanding of the context of these investigations.; Le Fundamentalsatz de Krull, généralisation du théorème de décomposition en facteurs premiers aux anneaux intègres, est le point de départ de la carrière de Lorenzen en mathématiques. Cet article retrace une histoire conceptuelle des reformulations successives par Lorenzen du Fundamentalsatz sur la base d'extraits de ses articles. Une édition de la correspondance connue de Lorenzen avec Hasse, Krull et Aubert permet une meilleure compréhension du contexte de ces recherches.

International audience; Krull's Fundamentalsatz, the generalisation of the main theorem of elementary number theory to integral domains, is the starting point of Lorenzen’s career in mathematics. This article traces a conceptual history of Lorenzen's successive reformulations of the Fundamentalsatz on the basis of excerpts of his articles. An edition of the extant correspondence of Lorenzen with Hasse, Krull, and Aubert provides a better understanding of the context of these investigations.; Le Fundamentalsatz de Krull, généralisation du théorème de décomposition en facteurs premiers aux anneaux intègres, est le point de départ de la carrière de Lorenzen en mathématiques. Cet article retrace une histoire conceptuelle des reformulations successives par Lorenzen du Fundamentalsatz sur la base d'extraits de ses articles. Une édition de la correspondance connue de Lorenzen avec Hasse, Krull et Aubert permet une meilleure compréhension du contexte de ces recherches.

We introduce the restricted local volume of a relatively very ample invertible sheaf as an invariant of equisingularity by determining its change across families. We apply this result to give numerical control of Whitney-Thom (differential) equisingularity for families of isolated complex analytic singularities. The characterization of the vanishing of the local volume gives rise to the class of deficient conormal (dc) singularities. We introduce a notion of generalized smoothability by considering the class of singularities that deform to dc singularities. Using Whitney stratifications and the functoriality properties of conormal spaces we show that fibers of conormal spaces are well-behaved under transverse maps. Then by Thom's transversality, the structure theorems of Hilbert-Burch and Buchsbaum-Eisenbud, we show that all smoothable singularities of dimension at least 2, Cohen-Macaulay codimension 2, Gorenstein codimension 3, and more generally determinantal and Pfaffian singularities deform to dc singularities., 45 pages; some results in Section 8 are strengthen by eliminating the isolated singularities hypothesis; corrected typos

International audience; In this chapter, we present methods to simplify reducible linear differential systems before solving. Classical integrals appear naturally as solutions of such systems. We will illustrate the methods developed in [DW19] on several examples to reduce the differential system. This will give information on potential algebraic relations between integrals.

28 pages. Dedicated to David Eisenbud on the occasion of his seventy-fifth birthday.; International audience; Parameterized algebraic curves and surfaces are widely used in geometric modeling and their manipulation is an important task in the processing of geometric models. In particular, the determination of the intersection loci between points, pieces of parameterized algebraic curves and pieces of algebraic surfaces is a key problem in this context. In this paper, we survey recent methods based on syzygies and blowup algebras for computing the image and the finite fibers of a curve or surface parameterization, more generally of a rational map. Conceptually, the main idea is to use elimination matrices, mainly built from syzygies, as representations of rational maps and to extract geometric informations from them. The construction and main properties of these matrices are first reviewed and then illustrated through several settings, each of them highlighting a particular feature of this approach that combines tools from commutative algebra, algebraic geometric and elimination theory.

We develop the algebraic formalism of the formal ternary laws of C. Walter and we compare them to Buchstaber's 2-valued formal group laws. We also compute the "elementary" formal ternary laws (after inverting 2) using a computer program available online., 35 pages. Updated introduction and various typos fixed. Comments are still welcome!

International audience; Refinement of neutrosophic algebraic structure or hyperstructure allows for the splitting of the indeterminate factor into different sub-indeterminate and gives a detailed information about the neutrosophic structure/hyperstructure considered. This paper is concerned with the development of a refined neutrosophic canonical hypergroup from a canonical hypergroup R and sub-indeterminate I 1 and I 2. Several interesting results and examples are presented. The paper also studies refined neutrosophic homomorphisms and establishes the existence of a good homomorphism between a refined neutrosophic canonical hypergroup R(I 1 , I 2) and a neutrosophic canonical hypergroup R(I).

International audience

In these lectures, we provide a toolkit to work with Chow-Witt groups, and more generally with the homology and cohomology of the Rost-Schmid complex associated to Milnor-Witt $K$-theory., Comment: This is an almost final version of the lecture notes on Chow-Witt groups that are going to appear in the proceedings "Motivic homotopy theory and refined enumerative geometry"

We investigate the problem of representing moment sequences by measures in the context ofPolynomial Optimization Problems. This consists in finding the infimum of a real polynomial ona real semialgebraic set defined by polynomial inequalities. We analyze the exactness of MomentMatrix (MoM) relaxations, dual to the Sum of Squares (SoS) relaxations, which are hierarchiesof convex cones introduced by Lasserre to approximate measures and positive polynomials. Weinvestigate in particular flat truncation properties, which allow testing effectively when MoMexactness holds.We consider the quadratic module Q generated by the inequalities. We show that the dualof the MoM relaxation coincides with the SoS relaxation extended with the real radical ofthe support of Q, and focus on the zero-dimensional case, generalizing results for equationsdefining a finite real variety. We deduce sufficient and necessary conditions for flat truncation,under the finite convergence assumption: flat truncation happens if and only if the support ofthe quadratic module associated with the minimizers is of dimension zero. We also bound theorder of the relaxation at which flat truncation holds.As corollaries, we conclude that flat truncation holds:$\bullet$ when regularity conditions, known as Boundary Hessian Conditions, hold: this resultimplies that flat truncation and MoM exactness holds generically;$\bullet$ when the support of the quadratic module Q is zero-dimensional;$\bullet$ in singular cases, flat truncation holds for the MoM relaxation extended with the polarconstraints when the real variety of polar points is finite.Effective numerical computations illustrate these flat truncation properties.

Affine ind-varieties are infinite dimensional generalizations of algebraic varieties which appear naturally in many different contexts, in particular in the study of automorphism groups of affine spaces. In this article we introduce and develop the basic algebraic theory of topologically integrable derivations of complete topological rings. We establish a bijective algebro-geometric correspondence between additive group actions on affine ind-varieties and topologically integrable derivations of their coordinate pro-rings which extends the classical fruitful correspondence between additive group actions on affine varieties and locally nilpotent derivations of their coordinate rings.

The lecture notes below correspond to the course given by the author in occasion of the VIASM school on Number Theory (18–24 June 2018, Hanoi). We have chosen to omit the proofs that are already presented in details in many references in the literature, although they were explained during the lectures, and we have devoted more space to statements useful in the applications. The applications concern many different mathematical settings, where linear difference equations naturally arise. We cite in particular the case of Drinfeld modules, which is considered in [Pel] and [TR].

The best known asymptotic bit complexity bound for factoring univariate polynomials over finite fields grows with the input degree to a power close to 1.5, and with the square of the bitsize of the ground field. It relies on a variant of the Cantor-Zassenhaus algorithm which exploits fast modular composition. Using techniques by Kaltofen and Shoup, we prove a refinement of this bound when the finite field has a large extension degree over its prime field. We also present fast practical algorithms for the case when the extension degree is smooth.

To appear in : Trans. Amer. Math. Soc.; We investigate the asymptotic behaviour of Castelnuovo-Mumford regularity of Ext and Tor, with respect to the homological degree, over complete intersection rings. We derive from a theorem of Gulliksen a linearity result for the regularity of Ext modules in high homological degrees. We show a similar result for Tor, under the additional hypothesis that high enough Tor modules are supported in dimension at most one; we then provide examples showing that the behaviour could be pretty hectic when the latter condition is not satisfied.

To appear in : Trans. Amer. Math. Soc.; We investigate the asymptotic behaviour of Castelnuovo-Mumford regularity of Ext and Tor, with respect to the homological degree, over complete intersection rings. We derive from a theorem of Gulliksen a linearity result for the regularity of Ext modules in high homological degrees. We show a similar result for Tor, under the additional hypothesis that high enough Tor modules are supported in dimension at most one; we then provide examples showing that the behaviour could be pretty hectic when the latter condition is not satisfied.

The main focus is the generic freeness of local cohomology modules in a graded setting. The present approach takes place in a quite nonrestrictive setting, by solely assuming that the ground coefficient ring is Noetherian. Under additional assumptions, such as when the latter is reduced or a domain, the outcome turns out to be stronger. One important application of these considerations is to the specialization of rational maps and of symmetric and Rees powers of a module., Comment: To appear in Transactions of the American Mathematical Society

Let K be a commutative field, M n (K) the K-algebra of n × nmatrices with coefficients in K. We show that for a subspace V of M n (K) withelements commuting two by two its dimension is bounded above by 1 + α(n)where α(n) is equal to r 2 for n = 2r and to r(r + 1) for n = 2r + 1. We alsodescribe those for which the dimension is 1 + α(n). This leads to an upperbound for the dimension of the commutative subalgebras of M n (K) and to thedescription of these for which the dimension is 1 + α(n). Then we deduce acharacterization of the maximal commutative subgroups of the linear groupGL n (K) and also of its maximal unipotent subgroups. The main interest in thispaper is that the proofs use only classical linear algebra.Mots-clés: Matrice, partie commutative, matrice nilpotente, matrice unipo-tente, algèbre commutative de matrices, groupe commutatif de matrices.; Soient K un corps commutatif, M n (K) la K-algèbre des matricesà n lignes et n colonnes à coefficients dans K. On montre ici que si V est unsous-espace vectoriel de M n (K) constitué de matrices qui commutent deux àdeux, alors la dimension de V est majorée par 1 + α(n) où α(n) est défini par r 2si n = 2r ou par r(r + 1) si n = 2r + 1. De plus on peut décrire les sous-espacesvectoriels constitués de matrices qui commutent deux à deux et de dimension1 + α(n). Cela nous conduit à majorer la dimension des sous-algèbres commuta-tives de M n (K) et à la description des sous-algèbres commutatives qui sont dedimension 1 + α(n). Il s’ensuit une caractérisation des sous-groupes commutat-ifs maximaux de GL n (K), ainsi que des sous-groupes unipotents maximaux deGL n (K). L’intérêt de cet article est que les démonstrations utilisent seulementles résultats classiques de l’algèbre linéaire.

In 2011, Khurana, Lam and Wang define the following property. (*)A commutative unital ring A satisfies the property ''power stable range one'' if for all a, b $\in$ A with aA + bA = A there are an integer N = N (a, b) $\ge$ 1 and $\lambda$ = $\lambda$(a, b) $\in$ A such that b N + $\lambda$a $\in$ A x , the unit group of A. In 2019, Berman and Erman consider rings with the following property (**) A commutative unital ring A has enough homogeneous polynomials if for any k $\ge$ 1 and set S := {p 1 , p 2 , ..., p k } , of primitive points in A n and any n $\ge$ 2, there exists an homogeneous polynomial P (X 1 , X 2 , ..., X n) $\in$ A[X 1 , X 2 , ..., X n ]) with deg P $\ge$ 1 and P (p i) $\in$ A x for 1 $\le$ i $\le$ k. We show in this article that the two properties (*) and (**) are equivalent and we shall call a commutative unital ring with these properties a good ring. When A is a commutative unital ring of pictorsion as defined by Gabber, Lorenzini and Liu in 2015, we show that A is a good ring. Using a Dedekind domain we built by Goldman in 1963,we show that the converse is false., Comment: The paper is reorganized, some historical comments are added and some proofs are streamlined following referee's suggestions

Let K be a field with a valuation $\nu$ and let L = K(x) be a transcendental extension of K, then any valuation $\mu$ of L which extends $\nu$ is determined by its restriction to the polynomial ring K[x]. We know how to associate to this valuation $\mu$ a family of valuations A = ($\mu$i)i$\in$I of K[x], called the associated admise family, which converges in a certain sense towards the valuation $\mu$. Although the definition of this family, as well as the notion of convergence, essentially imply the structure of the polynomial ring, in particular the degree of polynomials, we show in this note that the family A of valuations of L do not depend on the chosen generator x., Comment: in French

The purpose of this thesis is to study the questions surrounding the branching problem in representation theory using algebraic and combinatorial methods. Based on previously built models and ideas of the others, we develop and create new techniques, models to achieve deeper results. Concretely, we focus on two main projects: In the first project, we study the branching problem of affine Kac-Moody algebras g on its winding subalgebras g[u] by algebraic methods. Let h be the Cartan subalgebra of h. We prove that the support Gamma(g,g[u]) and Gamma(g,h) of the decomposition g-modules as g[u]-modules and h-modules are semigroups. Let Lambda, lambda be dominant integral weights of g, g[u], respectively. In the cases where g are types A_1^(1) and A_2^(2), with a certain condition on lambda, we can describe the set of numbers b such that (Lambda, lamba + b delta) are in Gamma(g,g[u]). The result helps us to realize the relation between Gamma(g,g[u]) and its saturated setting. In the second project, we study the coefficients appear in projective representation theory of symmetric groups: the shifted Littlewood-Richardson coefficients f_{lambda,mu}^nu (lambda, mu, nu are strict partitions) and g_{lambda,mu} (lambda is a strict partition and mu is partition) which can be considered as special cases of shifted Littlewood-Richardson coefficients. We obtain a new interpretation for the coefficients f_{lambda,mu}^nu. Moreover, for the coefficients g_{lambda,mu}, we also obtain another combinatorial description, which allows us to see the relations between g_{lambda,mu} with Littlewood-Richardson coefficients c_{mu^t,mu}^{lambda~}. Specifically, we prove that g_{lambda,mu} = g_{lambda,mu^t}, and c_{mu^t,mu}^{lambda~} is not less than g_{lambda,mu}. We conjecture that c_{mu^t,mu}^{lambda~} is not less than the square of g_{lambda,mu} and formulate some conjectures on our combinatorial models which would imply this inequality if it is valid, Le but de cette thèse est d'étudier les questions entourant le problème de branchement en théorie des représentations à l'aide de méthodes algébriques et combinatoires. Sur la base des modèles et des idées des autres construits précédemment, nous développons et créons de nouvelles techniques, des modèles pour obtenir des résultats plus profonds. Concrètement, nous nous concentrons sur deux projets principaux : Dans le premier projet, nous étudions le problème de branchement des algèbres Kac-Moody affines g sur ses sous-algèbres sinueuses g[u] par des méthodes algébriques. Soit h la sous-algèbre Cartan de g. Nous prouvons que le support Gamma(g,g[u]) et Gamma(g,h) de la décomposition de g-modules en tant que g[u]-modules et h-modules sont des semi-groupes. Soit Lambda, lambda des poids entiers dominants de g, g[u], respectivement. Soit delta la racine imaginaire de base de g. Dans les cas où g est de type A_1^(1) et A_2^(2), avec une certaine condition sur lambda, nous pouvons décrire l'ensemble des nombres b tels que (Lambda, lamba + b delta) sont dans Gamma(g,g[u]). Le résultat nous aide à réaliser la relation entre Gamma(g,g[u]) et son paramètre saturé. Dans le deuxième projet, nous étudions les coefficients apparaissant dans la théorie de représentation projective des groupes de symétrique: les coefficients décalés de Littlewood-Richardson f_{lambda,mu}^nu (lambda, mu, nu sont des partitions strictes) et g_{lambda, mu} (lambda est une partition stricte et mu est une partition) qui peuvent être considérés comme des cas particuliers de coefficients Littlewood-Richardson décalés. Nous obtenons une nouvelle interprétation pour les coefficients f_{lambda,mu}^nu. De plus, pour les coefficients g_{lambda,mu}, nous obtenons également une autre description combinatoire, qui nous permet de voir les relations entre g_{lambda,mu} avec les coefficients Littlewood-Richardson c_{mu^t,mu}^{lambda~}. Plus précisément, nous prouvons que g_{lambda,mu} = g_{lambda,mu^t}, et c_{mu^t,mu}^{lambda~} n’est pas moins que g_{lambda,mu^t}. Nous conjecturons que c_{mu^t,mu}^{lambda~} n’est pas moins que le carré de g_{lambda,mu^t} et formulons quelques conjectures sur nos modèles combinatoires qui impliquent cette inégalité si cela est valable

Let $\mathcal{C}$ be a plane curve given by an equation $f(x,y)=0$ with $f\in K[x][y]$ a monic squarefree polynomial. We study the problem of computing an integral basis of the algebraic function field $K(\mathcal{C})$ and give new complexity bounds for three known algorithms dealing with this problem. For each algorithm, we study its subroutines and, when it is possible, we modify or replace them so as to take advantage of faster primitives. Then, we combine complexity results to derive an overall complexity estimate for each algorithm. In particular, we modify an algorithm due to B\"ohm et al. and achieve a quasi-optimal runtime., Comment: Preliminary version

Peirce-evanescent baric identities are polynomial identities verified by baric algebras such that their Peirce polynomials are the null polynomial. In this paper procedures for constructing such homogeneous and non homogeneous identities are given. For this we define an algebraic system structure on the free commutative nonassociative algebra generated by a set T which provides for classes of baric algebras satisfying a given set of identities similar properties to those of the varieties of algebras. Rooted binary trees with labeled leaves are used to explain the Peirce polynomials. It is shown that the mutation algebras satisfy all Peirce-evanescent identities, it results from this that any part of the field K can be the Peirce spectrum of a K-algebra satisfying a Peirce-evanescent identity. We end by giving methods to obtain generators of homogeneous and non-homogeneous Peirce-evanescent identities that are applied in several univariate and multivariate cases.

One proves a far-reaching upper bound for the degree of a generically finite rational map between projective varieties over a base field of arbitrary characteristic. The bound is expressed as a product of certain degrees that appear naturally by considering the Rees algebra (blowup) of the base ideal defining the map. Several special cases are obtained as consequences, some of which cover and extend previous results in the literature., The last version to appear in the Journal of Algebra. Some changes have been made in the proof of the main theorem

Let K be a field equipped with a valuation. Tropical varieties over K can be defined with a theory of Gr{\"o}bner bases taking into account the valuation of K. Because of the use of the valuation, the theory of tropical Gr{\"o}bner bases has proved to provide settings for computations over polynomial rings over a p-adic field that are more stable than that of classical Gr{\"o}bner bases. In this article, we investigate how the FGLM change of ordering algorithm can be adapted to the tropical setting. As the valuations of the polynomial coefficients are taken into account, the classical FGLM algorithm's incremental way, monomo-mial by monomial, to compute the multiplication matrices and the change of basis matrix can not be transposed at all to the tropical setting. We mitigate this issue by developing new linear algebra algorithms and apply them to our new tropical FGLM algorithms. Motivations are twofold. Firstly, to compute tropical varieties, one usually goes through the computation of many tropical Gr{\"o}bner bases defined for varying weights (and then varying term orders). For an ideal of dimension 0, the tropical FGLM algorithm provides an efficient way to go from a tropical Gr{\"o}bner basis from one weight to one for another weight. Secondly, the FGLM strategy can be applied to go from a tropical Gr{\"o}bner basis to a classical Gr{\"o}bner basis. We provide tools to chain the stable computation of a tropical Gr{\"o}bner basis (for weight [0,. .. , 0]) with the p-adic stabilized variants of FGLM of [RV16] to compute a lexicographical or shape position basis. All our algorithms have been implemented into SageMath. We provide numerical examples to illustrate time-complexity. We then illustrate the superiority of our strategy regarding to the stability of p-adic numerical computations.

In this paper we provide a new method to certify that a nearby polynomial system has a singular isolated root with a prescribed multiplicity structure. More precisely, given a polynomial system f $=(f\_1, \ldots, f\_N)\in C[x\_1, \ldots, x\_n]^N$, we present a Newton iteration on an extended deflated system that locally converges, under regularity conditions, to a small deformation of $f$ such that this deformed system has an exact singular root. The iteration simultaneously converges to the coordinates of the singular root and the coefficients of the so called inverse system that describes the multiplicity structure at the root. We use $$\alpha$$-theory test to certify the quadratic convergence, and togive bounds on the size of the deformation and on the approximation error. The approach relies on an analysis of the punctual Hilbert scheme, for which we provide a new description. We show in particular that some of its strata can be rationally parametrized and exploit these parametrizations in the certification. We show in numerical experimentation how the approximate inverse system can be computed as a starting point of the Newton iterations and the fast numerical convergence to the singular root with its multiplicity structure, certified by our criteria., Comment: International Symposium on Symbolic and Algebraic Computation, Jul 2020, Kalamata, France

A topological commutative ring is said to be rigid when for every set $X$, the topological dual of the $X$-fold topological product of the ring is isomorphic to the free module over $X$. Examples are fields with a ring topology, discrete rings, and normed algebras. Rigidity translates into a dual equivalence between categories of free modules and of "topologically-free" modules and, with a suitable topological tensor product for the latter, one proves that it lifts to an equivalence between monoids in this category (some suitably generalized topological algebras) and coalgebras. In particular, we provide a description of its relationship with the standard duality between algebras and coalgebras, namely finite duality., This version only deals with the monoidalily of the equivalence

A tensor product surface $\mathscr{S}$ is an algebraic surface that is defined as the closure of the image of a rational map $\phi$ from $\mathbb{P}^1\times \mathbb{P}^1$ to $\mathbb{P}^3$. We provide new determinantal representations of $\mathscr{S}$ under the assumptions that $\phi$ is generically injective and its base points are finitely many and locally complete intersections. These determinantal representations are matrices that are built from the coefficients of linear relations (syzygies) and quadratic relations of the bihomogeneous polynomials defining $\phi$. Our approach relies on a formalization and generalization of the method of moving quadrics introduced and studied by David Cox and his co-authors., Comment: 18 pages. Revised version. Accepted for publication in Transactions of the AMS

13 pages; International audience; Generalizing the main result of (Aparicio, Compoint, Weil 2013), we prove that a system is in reduced form in the sense of Kochin and Kovacic if and only if any differential module in an algebraic construction admits a constant basis. Then we derive an explicit version of this statement, which was implicit in (Aparicio, Compoint, Weil 2013) and is a crucial ingredient of (Barkatou, Cluzeau, Di Vizio, Weil 2016). We finally deduce some properties of the Lie algebra of Katz's intrinsic Galois group, which is actually another fundamental ingredient of the algorithm in (Barkatou, Cluzeau, Di Vizio, Weil 2016).

The category of distributive lattices is, in classical mathematics, antiequivalent to the category of spectral spaces. We give here some examples and a short dictionary for this antiequivalence. We propose a translation of several abstract theorems (in classical mathematics) into constructive ones, even in the case where points of a spectral space have no clear constructive content.

Let $(K, \nu)$ be a valued field, the notions of \emph{augmented valuation}, of \emph{limit augmented valuation} and of \emph{admissible family} of valuations enable to give a description of any valuation $\mu$ of $K [x]$ extending $\nu$. In the case where the field $K$ is algebraically closed, this description is particularly simple and we can reduce it to the notions of \emph{minimal pair} and \emph{pseudo-convergent family}. Let $(K, \nu )$ be a henselian valued field and $\bar\nu$ the unique extension of $\nu$ to the algebraic closure $\bar K$ of $K$ and let $\mu$ be a valuation of $ K [x]$ extending $\nu$, we study the extensions $\bar\mu$ from $\mu$ to $\bar K [x]$ and we give a description of the valuations $\bar\mu_i$ of $\bar K [x]$ which are the extensions of the valuations $\mu_i$ belonging to the admissible family associated with $\mu$., Comment: in French

We give an example of a real algebraic manifold embedded in a complex space that does not satisfy the Nash-Artin approximation Property. This Nash-Artin approximation Property is closely related to the problem of determining when the biholomorphic equivalence for germs of real algebraic manifolds coincides with the algebraic equivalence. This example is an elliptic Bishop surface, and its construction is based on the functional equation satisfied by the generating series of some walks restricted to the quarter plane., to appear in Proceedings of the A.M.S

International audience; In [17] it was proved that if a differential field (K, δ) of charaacteris-tic 0 is algebraically closed and closed under Picard-Vessiot extensions then every differential algebraic P HS over K for a linear differential algebraic group G over K has a K-rational point (in fact if and only if). This paper explores whether and if so, how, this can be extended to (a) several commuting derivations, (b) one automorphism. Under a natural notion of "generalized Picard-Vessiot extension" (in the case of several derivations), we give a counterexample. We also have a counterexample in the case of one automorphism. We also formulate and prove some positive statements in the case of several derivations. On a montré dans [17] que si un corps différentiel (K, δ) de car-actéristique 0 est algébriquement clos et clos par extensions de Picard-Vessiot, alors tout espace principal homogène différentiel algébrique sur K a un point K-rationnel (et réciproquement). Cet article explore s'il est possible, et si oui comment, d'étendre ce résultat au cas de (a) plusieurs dérivations qui commutent, (b) un automorphisme. Pour une notion naturelle d'"extension de Picard-Vessiot généralisée" (dans le cas de plusieurs dérivations) nous donnons un contre-exemple. Nous avons aussi un contre-exemple dans le cas d'un automorphisme. Enfin, nous formulons et démontrons quelques résultats positifs dans le cas de plusieurs dérivations.

We give examples showing that the usual Artin Approximation theorems valid for convergent series over a field are no longer true for convergent series over a commutative Banach algebra. In particular we construct an example of a commutative integral Banach algebra $R$ such that the ring of formal power series over $R$ is not flat over the ring of convergent power series over $R$., To appear in Commun. Algebra

International audience; We study the WLP and SLP of artinian monomial ideals in S = K[x 1 ,. .. , x n ] via studying their minimal free resolutions. We study the Lefschetz properties of such ideals where the minimal free resolution of S/I is linear for at least n − 2 steps. We give an affirmative answer to a conjecture of Eisenbud, Huneke and Ulrich for artinian monomial ideals with almost linear resolutions.

International audience; Let $K$ be a field equipped with a valuation. Tropical varieties over $K$ can be defined with a theory of Gröbner bases taking into account the valuation of $K$. Because of the use of the valuation, this theory is promising for stable computations over polynomial rings over a $p$-adic fields.We design a strategy to compute such tropical Gröbner bases by adapting the Matrix-F5 algorithm. Two variants of the Matrix-F5 algorithm, depending on how the Macaulay matrices are built, are available to tropical computation with respective modifications. The former is more numerically stable while the latter is faster.Our study is performed both over any exact field with valuation and some inexact fields like $\mathbb{Q}_p$ or $\mathbb{F}_q \llbracket t \rrbracket.$ In the latter case, we track the loss in precision, and show that the numerical stability can compare very favorably to the case of classical Gröbner bases when the valuation is non-trivial. Numerical examples are provided.