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We prove that there exists a functorial correspondence between MV-algebras and partially cyclically ordered groups which are wound round of lattice-ordered groups. It follows that some results about cyclically ordered groups can be stated in terms of MV-algebras. For example, the study of groups together with a cyclic order allows to get a first-order characterization of groups of unimodular complex numbers and of finite cyclic groups. We deduce a characterization of pseudofinite MV-chains and of pseudo-simple MV-chains (i.e. which share the same first-order properties as some simple ones). We can generalize these results to some non-lineraly ordered MV-algebras, for example hyper-archimedean MV-algebras.

This article contains a complete proof of Gabrielov's rank Theorem, a fundamental result in the study of analytic map germs. Inspired by the works of Gabrielov and Tougeron, we develop formal-geometric techniques which clarify the difficult parts of the original proof. These techniques are of independent interest, and we illustrate this by adding a new (very short) proof of the Abhyankar-Jung Theorem. We include, furthermore, new extensions of the rank Theorem (concerning the Zariski main Theorem and elimination theory) to commutative algebra.

In Communication theory and Coding, it is expected that certain circulant matrices having $k$ ones and $k+1$ zeros in the first row are nonsingular. We prove that such matrices 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 construct circulant matrices having determinant $0$. The smallest singular matrix appears when $2k+1=45$. The possibility for such matrices to be singular is rather low, smaller than $10^{-4}$ in this case., Comment: 12 pages. To be published in Linear Algebra and Its Applications

We use an algebraic approach to construct minimal decompositions of symmetric tensors with low rank. This is done by using Apolarity Theory and by studying minimal sets of reduced points apolar to a given symmetric tensor, namely, whose ideal is contained in the apolar ideal associated to the tensor. In particular, we focus on the structure of the Hilbert function of these ideals of points. We give a procedure which produces a minimal set of points apolar to any symmetric tensor of rank at most 5. This procedure is also implemented in the algebra software Macaulay2., 26 pages; the Macaulay2 package "ApolarLowRank.m2" is available in the ancillary files

Soit $f:X\to S$ un morphisme surjectif et de pr\'esentation finie de sch\'emas int\`egres. Lorsque $S$ est de type fini sur un corps ou sur $\mathbb{Z}$, et de dimension $>0$, Poonen a montr\'e qu'il existe un point $x\in X$ dont le corps r\'esiduel $\kappa(x)$ est une extension radicielle de $\kappa(f(x))$. Dans ce travail, on prolonge ce r\'esultat de plusieurs fa\c{c}ons. D'abord, on peut choisir $x$ tel que $f(x)$ soit un point de codimension $1$ de $S$; si $S$ est lisse sur un corps $k$, on peut exiger que $\kappa(f(x))$ soit s\'eparable sur $k$. Dans une autre direction, on montre des r\'esultats analogues pour d'autres classes de sch\'emas $S$, par exemple les sch\'emas noeth\'eriens int\`egres de dimension $\geq2$. Let $f:X\to S$ be a morphism of integral schemes, which is surjective and of finite presentation. When $S$ is of finite type over a field or over $\mathbb{Z}$, of positive dimension, Poonen has shown that there is a point $x\in X$ whose residue field $\kappa(x)$ is purely inseparable over $\kappa(f(x))$. In this paper, we extend this result in several ways. First we prove that we can take $x$ such that $f(x)$ is a codimension $1$ point of $S$; if $S$ is smooth over a fiel $k$, we can require $\kappa(f(x))$ to be separable over $k$. In another direction, we prove that similar results hold for other schemes $S$, such as noetherian integral schemes of dimension $\geq2$., Comment: 15 pages, in French. Major revision, with new results. Accepted for publication in JTNB

In this paper, we study several definitions of generalized rank weights for arbitrary finite extensions of fields. We prove that all these definitions coincide, generalizing known results for extensions of finite fields., Comment: Comments welcome!

We refine necessary and sufficient conditions for the generating series of a weighted model of a quarter plane walk to be differentially algebraic. In addition, we give algorithms based on the theory of Mordell-Weil lattices, that, for each weighted model, yield polynomial conditions on the weights determining this property of the associated generating series., 37 Pages

International audience; This paper presents the relationship between differential algebra and tropical differential algebraic geometry, mostly focusing on the existence problem of formal power series solutions for systems of polynomial ODE and PDE. Moreover, it improves an approximation theorem involved in the proof of the Fundamental Theorem of tropical differential algebraic geometry which permits to improve this latter by dropping the base field uncountability hypothesis used in the original version.

We prove stability of logarithmic tangent sheaves of singular hypersurfaces D of the projective space with constraints on the dimension and degree of the singularities of D. As main application, we prove that determinants and symmetric determinants have stable logarithmic tangent sheaves and we describe an open dense piece of the associated moduli space., Major update. To appear in IMRN

International audience; We show a remarkable property of the CM-wild variety P-1 x P-2 , namely that the only ACM sheaves moving in positive-dimensional families are Ulrich bundles. A complete classification of the non-Ulrich range is given.We prove that this feature is unique in the sense that any other ACM reduced closed subscheme X subset of P-N of dimension n >= 1 belongs to the well-known list of CM-finite or CM-tame varieties, or else it remains CM-wild upon removing Ulrich sheaves.

International audience; Assuming the variety of a polynomial set is invariant under a group action, we construct a set of invariants that define the same variety. Our construction can be seen as a generalization of the previously known construction for finite groups. The result though has to be understood outside an invariant variety which is independent of the polynomial set considered. We introduce the symmetrizations of a polynomial that are polynomials in a generating set of rational invariants. The generating set of rational invariants and the symmetrizations are constructed w.r.t. a section to the orbits of the group action.

International audience; A Hilbertian (co)algebra is defined as a (co)semigroup object in the monoidal category of Hilbert spaces. The carrier Hilbert space of such an algebra splits as an orthogonal direct sum of its Jacobson radical and the closure of the linear span of a special class of elements, the group-like elements of its adjoint coalgebra, which by the Riesz representation, correspond to closed maximal modular ideals. When the coproduct is isometric, semisimplicity is shown to be equivalent to the existence of adjoints in the sense of Ambrose’s H*-algebras. We also prove that the category of semisimple special Hilbertian algebras, that is, semisimple Hilbertian algebras with an isometric coproduct, i.e., essentially the algebras of the form l2(X) with the pointwise product, are dually equivalent to a subcategory of pointed sets and base-point preserving maps.

International audience; 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.

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

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.

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.

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.

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

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"

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 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 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

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 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.

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.

Resume Dans cet article nous presentons un algorithme pour calculer la forme de Hermite d'une pseudo-matrice sur un domaine de Prufer. Ceci nous donne des preuves constructives des principaux resultats theoriques concernant les modules de presentation finie et projectifs de type fini sur les domaines de Prufer, et nous permet de discuter la resolution des systemes lineaires sur les domaines de Prufer. Nous generalisons ainsi la methodologie developpee par Henri Cohen pour les domaines de Dedekind. Nous presentons egalement des resultats concernant la reduction de Smith sur les domaines de Prufer de dimension ⩽1.

Given an inverse semigroup $S$ endowed with a partial action on a topological space $X$, we construct a groupoid of germs $S\ltimes X$ in a manner similar to Exel's groupoid of germs, and similarly a partial action of $S$ on an algebra $A$ induces a crossed product $A\rtimes S$. We then prove, in the setting of partial actions, that if $X$ is locally compact Hausdorff and zero-dimensional, then the Steinberg algebra of the groupoid of germs $S\ltimes X$ is isomorphic to the crossed product $A_R(X)\rtimes S$, where $A_R(X)$ is the Steinberg algebra of $X$. We also prove that the converse holds, that is, that under natural hypotheses, crossed products of the form $A_R(X)\rtimes S$ are Steinberg algebras of appropriate groupoids of germs of the form $S\ltimes X$. We introduce a new notion of topologically principal partial actions, which correspond to topologically principal groupoids of germs, and study orbit equivalence for these actions in terms of isomorphisms of the corresponding groupoids of germs. This generalizes previous work of the first-named author as well as from others, which dealt mostly with global actions of semigroups or partial actions of groups. We finish the article by comparing our notion of orbit equivalence of actions and orbit equivalence of graphs., 34 pages

This chapter is devoted to a survey of the historical background of Grobner bases for D-modules and linear rewriting theory largely developed in algebra throughout the twentieth century and to present deep relationships between them. Completion methods are the main streams for these computational theories. In the theory of Grobner bases, they were motivated by algorithmic problems in elimination theory such as computations in quotient polynomial rings modulo an ideal, manipulating algebraic equations, and computing Hilbert series. In rewriting theory, they were motivated by computation of normal forms and linear bases for algebras and computational problems in homological algebra.

We construct an explicit filtration of the ring of algebraic power series by finite dimensional constructible sets, measuring the complexity of these series. As an application, we give a bound on the dimension of the set of algebraic power series of bounded complexity lying on an algebraic variety defined over the field of power series., 12 pages

The purpose of this note is to correct a mistake in the article “A curve selection lemma in spaces of arcs and the image of the Nash map” Compositio Math. 142 (2006), 119–130. It is due to an overlooked hypothesis in the definition of generically stable subset of the space of arcs X∞ of a variety X defined over a perfect field k.

International audience; We study binomiality of the steady state ideals of chemical reaction networks. Considering rate constants as indeterminates, the concept of unconditional binomiality has been introduced and an algorithm based on linear algebra has been proposed in a recent work for reversible chemical reaction networks, which has a polynomial time complexity upper bound on the number of species and reactions. In this article, using a modified version of species--reaction graphs, we present an algorithm based on graph theory which performs by adding and deleting edges and changing the labels of the edges in order to test unconditional binomiality. We have implemented our graph theoretical algorithm as well as the linear algebra one in Maple and made experiments on biochemical models. Our experiments show that the performance of the graph theoretical approach is similar to or better than the linear algebra approach, while it is drastically faster than Groebner basis and quantifier elimination methods.

International audience; Sparse interpolation refers to the exact recovery of a function as a short linear combination of basis functions from a limited number of evaluations. For multivariate functions, the case of the monomial basis is well studied, as is now the basis of exponential functions. Beyond the multivariate Chebyshev polynomial obtained as tensor products of univariate Chebyshev polynomials, the theory of root systems allows to define a variety of generalized multivariate Chebyshev polynomials that have connections to topics such as Fourier analysis and representations of Lie algebras. We present a deterministic algorithm to recover a function that is the linear combination of at most r such polynomials from the knowledge of r and an explicitly bounded number of evaluations of this function.

Let A be a finite subset of an abelian group ( G , + ) . For h ∈ N , let h A = A + … + A denote the h-fold iterated sumset of A. If | A | ≥ 2 , understanding the behavior of the sequence of cardinalities | h A | is a fundamental problem in additive combinatorics. For instance, if | h A | is known, what can one say about | ( h − 1 ) A | and | ( h + 1 ) A | ? The current classical answer is given by | ( h − 1 ) A | ≥ | h A | ( h − 1 ) / h , a consequence of Plunnecke's inequality based on graph theory. We tackle here this problem with a completely new approach, namely by invoking Macaulay's classical 1927 theorem on the growth of Hilbert functions of standard graded algebras. With it, we first obtain demonstrably strong bounds on | h A | as h grows. Then, using a recent condensed version of Macaulay's theorem, we derive the above Plunnecke-based estimate and significantly improve it in the form | ( h − 1 ) A | ≥ θ ( x , h ) | h A | ( h − 1 ) / h for h ≥ 2 and some explicit factor θ ( x , h ) > 1 , where x ∈ R satisfies x ≥ h and | h A | = ( x h ) . Equivalently and more simply, | ( h − 1 ) A | ≥ h x | h A | . We show that θ ( x , h ) often exceeds 1.5 and even 2, and asymptotically tends to e ≈ 2.718 as x grows and h lies in a suitable range depending on x.

International audience; We are interested in the evolution operators defined on commutative and non-associative algebras when the characteristic of the scalar field is 2. We distinguish four types: nilpotent, quasi-constant, ultimately periodic, and plenary train operators. They are studied and classified for non-baric and for baric algebras.

International audience; We contribute a new algebraic method for computing the orthogonal projections of a point onto a rational algebraic surface embedded in the three dimensional projective space. This problem is first turned into the computation of the finite fibers of a generically finite dominant rational map: a congruence of normal lines to the rational surface. Then, an in-depth study of certain syzygy modules associated to such a congruence is presented and applied to build elimination matrices that provide universal representations of its finite fibers, under some genericity assumptions. These matrices depend linearly in the variables of the three dimensional space. They can be pre-computed so that the orthogonal projections of points are approximately computed by means of fast and robust numerical linear algebra calculations.

International audience; Linear codes over finite rings are described here as trace codes for a suitable generalization of the trace called a GF-trace. Cyclic codes over Galois rings are given a trace description as well. The main tools are the notion of trace dual bases, in the case of linear codes, and of normal bases of an extension ring over a ring, in the case of cyclic codes.

Given a parameterization $\phi$ of a rational plane curve C, we study some invariants of C via $\phi$. We first focus on the characterization of rational cuspidal curves, in particular we establish a relation between the discriminant of the pull-back of a line via $\phi$, the dual curve of C and its singular points. Then, by analyzing the pull-backs of the global differential forms via $\phi$, we prove that the (nearly) freeness of a rational curve can be tested by inspecting the Hilbert function of the kernel of a canonical map. As a by product, we also show that the global Tjurina number of a rational curve can be computed directly from one of its parameterization, without relying on the computation of an equation of C., Comment: Mathematics of Computation, American Mathematical Society, In press

Let k be an infinite field and I ⊂ k [ x 1 , … , x n ] be a non-zero ideal such that dim V ( I ) = q ≥ 0 . Denote by ( f 1 , … , f s ) a set of generators of I. One can see that in the set I ∩ k [ x 1 , . . . , x q + 1 ] there exist non-zero polynomials, depending only on these q + 1 variables. We aim to bound the minimal degree of the polynomials of this type, and of a Bezout (i.e. membership) relation expressing such a polynomial as a combination of the f i . In particular we show that if deg ⁡ f i = d i where d 1 ≥ d 2 . . . ≥ d s , then there exist a non-zero polynomial ϕ ( x ) ∈ k [ x 1 , . . . , x q + 1 ] ∩ I , such that deg ⁡ ϕ ≤ d s ∏ i = 1 n − q − 1 d i . We also give a relative version of this theorem.

International audience; Sylvester doubles sums, introduced first by Sylvester (see Sylvester (1840, 1853)), are symmetric expressions of the roots of two polynomials. Sylvester’s definition of double sums makes no sense in the presence of multiple roots, since the definition involves denominators that vanish when there are multiple roots. The aim of this paper is to give a new definition of Sylvester double sums making sense in the presence of multiple roots, which coincides with the definition by Sylvester in the case of simple roots, to prove that double sums indexed by (a, b) are equal up to a constant if they share the same value for a+b, as well a proof of the relationship between double sums and subresultants, i.e. that they are equal up to a constant. In the simple root case, proofs of these properties are already known (see Lascoux and Pragacz (2002); d’Andrea et al. (2007); Roy and Szpirglas (2011)). The more general proofs given here are using generalized Vandermonde determinants and symmetric multivariate Hermite interpolation as well as an induction on the length of the remainder sequence of P and Q.

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.