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International audience; We address the description of the tropicalization of families of rational varieties under parametrizations with prescribed support, via curve valuations. We recover and extend results by Sturmfels, Tevelev and Yu for generic coefficients, considering rational parametrizations with non-trivial denominator. The advantage of our point of view is that it can be generalized to deal with non-generic parametrizations. We provide a detailed analysis of the degree of the closed image, based on combinatorial conditions on the relative positions of the supports of the polynomials defining the parametrization. We obtain a new formula and finer bounds on the degree, when the supports of the polynomials are different. We also present a new formula and bounds for the order at the origin in case the closed image is a hypersurface.

We study the form of possible algebraic relations between functions satisfying linear differential equations. In particular , if f and g satisfy linear differential equations and are algebraically dependent, we give conditions on the differential Galois group associated to f guaranteeing that g is a polynomial in f. We apply this to hypergeometric functions and iterated integrals.

International audience

We describe a recursive algorithm that decomposes an algebraic set into locally closed equidimensional sets, i.e. sets which each have irreducible components of the same dimension. At the core of this algorithm, we combine ideas from the theory of triangular sets, a.k.a. regular chains, with Gr\"obner bases to encode and work with locally closed algebraic sets. Equipped with this, our algorithm avoids projections of the algebraic sets that are decomposed and certain genericity assumptions frequently made when decomposing polynomial systems, such as assumptions about Noether position. This makes it produce fine decompositions on more structured systems where ensuring genericity assumptions often destroys the structure of the system at hand. Practical experiments demonstrate its efficiency compared to state-of-the-art implementations., Comment: Some minor revisions, corrects a mistake in the proof of lemma 2.2

A tri-linear rational map in dimension three is a rational map $\phi: (\mathbb{P}_\mathbb{C}^1)^3 \dashrightarrow \mathbb{P}_\mathbb{C}^3$ defined by four tri-linear polynomials without a common factor. If $\phi$ admits an inverse rational map $\phi^{-1}$, it is a tri-linear birational map. In this paper, we address computational and geometric aspects about these transformations. We give a characterization of birationality based on the first syzygies of the entries. More generally, we describe all the possible minimal graded free resolutions of the ideal generated by these entries. With respect to geometry, we show that the set $\mathfrak{Bir}_{(1,1,1)}$ of tri-linear birational maps, up to composition with an automorphism of $\mathbb{P}_\mathbb{C}^3$, is a locally closed algebraic subset of the Grassmannian of $4$-dimensional subspaces in the vector space of tri-linear polynomials, and has eight irreducible components. Additionally, the group action on $\mathfrak{Bir}_{(1,1,1)}$ given by composition with automorphisms of $(\mathbb{P}_\mathbb{C}^1)^3$ defines 19 orbits, and each of these orbits determines an isomorphism class of the base loci of these transformations., Comment: 28 pages, 1 figure; to appear in Mathematics of Computation (AMS)

The Casas-Alvero conjecture predicts that every univariate polynomial over a field of characteristic zero having a common factor with each of its derivatives $H_i (f )$ is a power of a linear polynomial. One approach to proving the conjecture is to first prove it for polynomials of some small degree $d$, compile a list of bad primes for that degree (namely, those primes p for which the conjecture fails in degree $d$ and characteristic $p$) and then deduce the conjecture for all degrees of the form $dp^\ell$ , $\ell \in \mathbb{N}$, where $p$ is a good prime for $d$. In this paper we calculate certain distinguished monomials appearing in the resultant $R(f, H_i (f ))$. As a corollary, we obtain a (non-exhaustive) list of bad primes for every degree $d \in \mathbb{N} \setminus 0$.

International audience; In this paper, we present a basic theory of the duality of linear codes over three of the non-unital rings of order four; namely I, E , and H as denoted in (Fine, 1993). A new notion of duality is introduced in the case of the non-commutative ring E . The notion of self-dual codes with respect to this duality coincides with that of quasi self-dual codes over E as introduced in (Alahmadi et al , 2022). We characterize self-dual codes and LCD codes over the three rings, and investigate the properties of their corresponding additive codes over F 4 . We study the connection between the dual of any linear code over these rings and the dual of its associated binary codes. A MacWilliams formula is established for linear codes over E .

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.

The representation of positive polynomials on a semi-algebraic set in terms of sums of squares is a central question in real algebraic geometry, which the Positivstellensatz answers. In this paper, we study the effective Putinar's Positivestellensatz on a compact basic semialgebraic set $S$ and provide a new proof and new improved bounds on the degree of the representation of positive polynomials. These new bounds involve a parameter $\epsilon$ measuring the non-vanishing of the positive function, the constant $c$ and exponent $L$ of a {\L}ojasiewicz inequality for the semi-algebraic distance function associated to the inequalities $g = (g_1 ,. .. , g_r)$ defining $S$. They are polynomial in $c$ and $\epsilon^{-1}$ with an exponent depending only on $L$. We analyse in details the {\L}ojasiewicz inequality when the defining inequalties g satisfy the Constraint Qualification Condition. We show that, in this case, the {\L}ojasiewicz exponent $L$ is 1 and we relate the {\L}ojasiewicz constant $c$ with the distance of $g$ to the set of singular systems.

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.

Let $k$ be a perfect field of characteristic $p > 0$, and let $K = k((u))$ be the field of Laurent series over $K$. We study the skew polynomial ring $K[T, \Phi]$, where $\Phi$ is an endomorphism of $K$ that extends a Frobenius endomorphism of $k$. We give a description of the irreducible skew polynomials, develop an analogue of the theory of the Newton polygon in this context, and classify the similarity classes of irreducible elements.

We analyse the representation of positive polynomials in terms of Sums of Squares. We provide a quantitative version of Putinar's Positivstellensatz over a compact basic semialgebraic set S, with a new polynomial bound on the degree of the positivity certificates. This bound involves a Lojasiewicz exponent associated to the description of S. We show that if the gradients of the active constraints are linearly independent on S (Constraint Qualification condition),this Lojasiewicz exponent is equal to 1. We deduce the first general polynomial bound on the convergence rate of the optima in Lasserre's Sum-of-Squares hierarchy to the global optimum of a polynomial function on S, and the first general bound on the Hausdorff distance between the cone of truncated (probability) measures supported on S and the cone of truncated pseudo-moment sequences, which are positive on the quadratic module of S., Comment: Mathematical Programming, Series A, 2022

International audience; Determinantal polynomial systems are those involving maximal minors of some given matrix. An important situation where these arise is the computation of the critical values of a polynomial map restricted to an algebraic set. This leads directly to a strategy for, among other problems, polynomial optimisation. Computing Gröbner bases is a classical method for solving polynomial systems in general. For practical computations, this consists of two main stages. First, a Gröbner basis is computedwith respect to a DRL (degree reverse lexicographic) ordering. Then, a change of ordering algorithm, such as \textsf{Sparse-FGLM}, designed by Faug\`ere and Mou, is used to find a Gröbner basis of the same system but with respect to a lexicographic ordering. The complexity of this latter step, in terms of the number of arithmetic operations in the ground field, is $O(mD^2)$, where $D$ is the degree of the ideal generated by the input and $m$ is the number of non-trivial columns of a certain $D \times D$ matrix. While asymptotic estimates are known for $m$ in the case of \textit{generic} polynomial systems, thus far, the complexity of \textsf{Sparse-FGLM} was unknown for the class of determinantal systems. By assuming Fröberg's conjecture, thus ensuring that the Hilbert series of generic determinantal ideals have the necessary structure, we expand the work of Moreno-Soc\'ias by detailing the structure of the DRL staircase in the determinantal setting. Then we study the asymptotics of the quantity $m$ by relating it to the coefficients of these Hilbert series. Consequently, we arrive at a new bound on the complexity of the \textsf{Sparse-FGLM} algorithm for generic determinantal systems and, in particular, for generic critical point systems. We consider the ideal inside the polynomial ring $\mathbb{K}[x_1, \dots, x_n]$, where $\mathbb{K}$ is some infinite field, generated by $p$ generic polynomials of degree $d$ and the maximal minors of a $p \times (n-1)$ polynomial matrix with generic entries of degree $d-1$. Then, in this setting, for the case $d=2$ and for $n \gg p$ we establish an exact formula for $m$ in terms of $n$ and $p$. Moreover, for $d \geq 3$, we give a tight asymptotic formula, as $n \to \infty$, for $m$ in terms of $n,p$ and $d$.

The derived bracket of a Maurer-Cartan element in a differential graded Lie algebra (DGLA) is well-known to define a differential graded Leibniz algebra. It is also well-known that a Lie infinity morphism between DGLAs maps a Maurer-Cartan element to a Maurer-Cartan element. Given a Lie-infinity morphism, a Maurer-element and its image, we show that both derived differential graded Leibniz algebras are related by a Leibniz-infinity morphism, and we construct it explicitely. As an application, we recover a well-known formula of Dominique Manchon about the commutator of the star-product. Mots-clefs : Algèbres de Leibniz, algèbre de Lie-infinies, formalité et quantification., Il est connu que le crochet dérivé d'un élément de Maurer-Cartan d'une algèbre de Lie différentielle graduée (DGLA) définit une algèbre de Leibniz différentielle graduée. Il est connu aussi que un morphisme de Lie infini entre DGLAs envoie un élément de Maurer-Cartan sur un autre élément de Maurer-Cartan. ´ Etant donnés un morphisme Lie-infini, un élément de Maurer-Cartan et son image, nous construisons entre leurs algèbres de Leibniz différentielles graduées un morphisme de Leibniz infini, et ce de façon totalement explicite. Nous utilisons cette construction pour retrouver une formule de Dominique Manchon à propos du commutateur du produit-étoile.

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

\'Etant donn\'e un anneau de valuation $V$, de corps r\'esiduel $F$ et de groupe des valeurs $\Gamma$, on donne une condition suffisante pour qu'un anneau local dominant $V$ soit un anneau de valuation de groupe $\Gamma$. Lorsque $V$ contient un corps $k$, ce r\'esultat est appliqu\'e \`a la construction d'un anneau de valuation contenant $V$ et une extension donn\'ee $k'$ de $k$, de groupe $\Gamma$ et de corps r\'esiduel engendr\'e par $k'$ et $F$. Cela s'av\`ere possible, notamment, lorsque $k'$ ou $F$ est s\'eparable sur $k$. Given a valuation ring $V$, with residue field $F$ and value group $\Gamma$, we give a sufficient condition for a local ring dominating $V$ to be a valuation ring with the same value group. When $V$ contains a field $k$, we apply this result to the problem of constructing a valuation ring $W$ containing $V$ and a prescribed extension $k'$ of $k$, with value group $\Gamma$ and residue field generated by $k'$ and $F$; this is possible in particular when either $k'$ or $F$ is separable over $k$., Comment: 11 pages, in French. Published in Rend. Sem. Mat. Univ. Padova (Online first)

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

The existence of certain Fq-spaces of differential forms of the projective line over a field K containing Fq leads us to prove an identity linking the determinant of the Moore matrix of n indeterminates with the determinant of the Moore matrix of the cofactors of its first row. These same spaces give an interpretation of Elkies pairing in terms of residues of differential forms. This pairing puts in duality the Fq-vector space of the roots of a Fq-linear polynomial and that of the roots of its reversed polynomial.

In polynomial optimization, two different and dual approaches are considered: the approximation of positive polynomials using sums of squares (SoS), that translates into Lasserre's SoS hierarchy, and the approximation of measures using truncated positive linear functionals (or truncated pseudo-moment sequences), that translates into Lasserre's moment hierarchy. In this thesis, we investigate exact and approximate representation properties in both the two cases. The representation of positive polynomials in terms of sums of squares is a central question in real algebraic geometry, that is answered by the Positivstellensätze. In particular, we investigate effective version of Putinar's Positivstellensatz, and provide new bounds on the degree of representation of a strictly positive polynomial on a basic compact semialgebraic set S, under the Archimedean condition. These bounds involve a parameter ε, measuring how close is the strictly positive polynomial to have a zero on the semialgebraic set: these are the first bounds with a polynomial dependence on ε -1. The bounds also show a new explicit dependence on the Łojasiewicz exponent Ł and constant cst, arising from a Łojasiewicz inequality between the distance and semialgebraic distance functions from S. We analyze in detail regular cases, where we can show that the Łojasiewicz exponent is equal to one and we have explicit bounds for the Łojasiewicz constant. We interpret our effective Putinar's Positivstellensatz as a result of quantitative approximation of positive polynomials, and deduce the first general polynomial convergence for Lasserre's hierarchies. On the dual side, we deduce convergence rates of truncated positive linear functionals (or truncated pseudo-moment sequances) to measures. We then move to exact representations on the dual side. We investigate properties of the dual cones of truncated quadratic modules, and we introduce the concept of exactness for Lasserre's moment hierarchy, that is closely related to the flat truncation property. We show that the dual of the moment hierarchy coincides with an extended SoS hierarchy, and we detail the analysis for zero dimensional semialgebraic sets. Finally, we apply the results obtained to the study of flat truncation. We give the first necessary and sufficient condition for flat truncation, under the finite convergence assumption, giving bounds for the order of relaxation needed. As corollaries, we conclude that flat truncation holds under generic assumptions, and we give a unified presentation of different results in the zero dimensional case. As an application, we present a new algorithm for computing the real radical of an ideal I. We exploit properties of truncated positive linear functionals and techniques from numerical algebraic geometry. We give an effective, general stopping criterion on the degree, to detect when the kernel of the moment matrix of a generic linear functional can be used to compute equations for the irreducible components of the real variety defined by I. Finally, we compute the real radical as the intersection of real prime ideals lying over I, and illustrate this approach in several examples., Dans le domaine de l'optimisation polynomiale, deux approches différentes et duales sont considérées : l'approximation de polynômes positifs à l'aide de sommes de carrés (SoS), qui se traduit par la hiérarchie SoS de Lasserre, et l'approximation de mesures à l'aide de fonctions linéaires positives tronquées (ou de séquences de pseudo-moments tronquées), qui se traduit par la hiérarchie des moments de Lasserre. Dans cette thèse, nous étudions les propriétés de représentation exactes et approchées dans les deux cas. La représentation des polynômes positifs en termes de sommes de carrés est une question centrale en géométrie algébrique réelle, à laquelle répondent les Positivstellensätze. En particulier, nous étudions une version effective du Positivstellensatz de Putinar, et fournissons de nouvelles bornes sur le degré de représentation d'un polynôme strictement positif sur un ensemble semialgébrique de base S compact, sous la condition Archimedienne. Ces bornes font intervenir un paramètre ε, qui mesure à quelle distance se trouve le polynôme strictement positif d'avoir un zéro sur l'ensemble semialgébrique : ce sont les premières bornes avec une dépendance polynomiale de ε -1. Dans les bornes, on trouve également une nouvelle dépendance explicite de l'exposant de Łojasiewicz Ł et de la constante cst, provenant d'une inégalité Łojasiewicz entre les fonctions de distance et de distance semialgébrique de S. Nous analysons en détail les cas réguliers, dans lesquels nous pouvons montrer que l'exposant Łojasiewicz est égal à un et nous avons des limites explicites pour la constante Łojasiewicz. Nous interprétons notre Positivstellensatz effectif de Putinar comme un résultat d'approximation quantitatif des polynômes positifs, et déduisons la première convergence polynomiale générale pour les hiérarchies de Lasserre. Du point de vue dual, nous déduisons les taux de convergence des fonctions linéaires positives tronquées (ou des séquences pseudo-momentales tronquées) vers les mesures. Nous passons ensuite aux représentations exactes dans le dual. Nous étudions les propriétés des cônes duaux des modules quadratiques tronqués, et nous introduisons le concept d'exactitude pour la hiérarchie des moments de Lasserre, qui est étroitement lié à la propriété de troncation plate. Nous montrons que le dual de la hiérarchie des moments coïncide avec une hiérarchie SoS étendue, et nous détaillons l'analyse pour les ensembles semialgébriques de dimension zéro. Enfin, nous appliquons les résultats obtenus à l'étude de la troncation plate. Nous donnons la première condition nécessaire et suffisante pour la troncature plate, sous l'hypothèse de convergence finie, en donnant des limites pour l'ordre de relaxation nécessaire. Comme corollaires, nous concluons que la troncation plate est vérifiée sous des hypothèses génériques, et nous donnons une présentation unifiée des différents résultats dans le cas de la dimension zéro. Comme application, nous présentons un nouvel algorithme pour calculer le radical réel d'un idéal I. Nous exploitons les propriétés des fonctions linéaires positives tronquées et les techniques de la géométrie algébrique numérique. Nous donnons un critère d'arrêt efficace et général sur le degré, pour détecter quand le noyau de la matrice des moments d'une fonction linéaire générique peut être utilisé pour calculer les équations des composantes irréductibles de la variété réelle définie par I. Enfin, nous calculons le radical réel comme l'intersection d'idéaux premiers réels contenant sur I, et illustrons cette approche par plusieurs exemples.

International audience; We prove an analogue of the Weierstrass preparation theorem for henselian pairs, generalizing the local case recently proved by Bouthier and Česnavičius. As an application, we construct a henselian analogue of the resultant of p-adic series defined by Berger.

This paper is concerned with linear algebra based methods for solving exactly polynomial systems through so-called Gr\"obner bases, which allow one to compute modulo the polynomial ideal generated by the input equations. This is a topical issue in non-linear algebra and more broadly in computational mathematics because of its numerous applications in engineering and computing sciences. Such applications often require geometric computing features such as representing the closure of the set difference of two solution sets to given polynomial systems. Algebraically, this boils down to computing Gr\"obner bases of colon and/or saturation polynomial ideals. In this paper, we describe and analyze new Gr\"obner bases algorithms for this task and present implementations which are more efficient by several orders of magnitude than the state-of-the-art software.

Vibration analysis aims at identifying potential failures of a rotating machinery from the monitoring of its vibration levels, i.e., by measuring the vibrations and comparing them to known failure vibration signals. For the diagnostic of gearboxes, new demodulation methods have recently been introduced in acoustic and signal processing. This new approach yields the problem of writing/factorizing a matrix M as D_1 u v_1 + ... + D_r u v_r = (D_1 u ... D_r u) (v_1^T ... v_r^T)^T,where the D_i's are fixed matrices, u (resp., v_i) is a row (resp., column) vector to be determined and i=1, ..., r. In this paper, using module theory and homological algebra, we study this rank factorization problem. More precisely, we characterize the general solutions of this family of polynomial systems. Finally, the results we develop are effective in the sense of computer algebra. Thus, they can be implemented in standard computer algebra systems handling polynomial systems and basic homological algebra methods (e.g., the Singular system, the GAP library CapAndHomalg, the Maple package OreModules).; L'analyse vibratoire a pour but d'identifier de potentiels défauts d'une machine tournante grâce à la surveillance de ses niveaux de vibration, c'est-à-dire, grâce à la mesure de ses vibrations et à leur comparaison avec des signaux de défauts connus. Pour le diagnostic d'engrenages, de nouvelles méthodes de démodulation ont récemment été introduites en acoustique et en traitement du signal. Cette nouvelle approche a permis l'étude du problème consistant à écrire/factoriser une matrice M sous la forme de D_1 u v_1 + ... + D_r u v_r = (D_1 u ... D_r u) (v_1^T ... v_r^T)^T, où les D_i sont des matrices fixées, u (resp., v_i) est un vecteur colonne (resp., un vecteur ligne) à déterminer et i=1, ..., r. Dans ce papier, en utilisant la théorie des modules et l'algèbre homologique, nous étudions ce problème de factorisation relative au rang. Plus précisément, nous caractérisons les solutions générales de cette famille de systèmes polynomiaux. Finalement, les résultats obtenus sont effectifs au sens du calcul formel. Ainsi, ils peuvent être implantés dans des systèmes standards de calcul formel permettant l'étude effective des systèmes polynomiaux et des méthodes élémentaires d'algèbre homologique (par exemple, le système Singular, la librairie CapAndHomalg de GAP, le package OreModules écrit en Maple).

We prove that stably isomorphic vector bundles of rank d-1 on a smooth affine d-fold X over an algebraically closed field k are indeed isomorphic, provided d! is invertible in k. This answers an old conjecture of Suslin., Comment: 18 pages. Preliminary version. Comments are welcome

We survey some recent progress in the theory of vector bundles on algebraic varieties and related questions in algebraic K-theory., 20 pages; ICM 2022 Proceedings

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 prove that exterior powers of (skew-)symmetric bundles induce a $\lambda$-ring structure on the ring $GW^0(X) \oplus GW^2(X)$, when $X$ is a scheme where $2$ is invertible. Using this structure, we define stable Adams operations on Hermitian $K$-theory. As a byproduct of our methods, we also compute the ternary laws associated to Hermitian $K$-theory.

Let $\iota:K\hookrightarrow L\cong K(x)$ be a simple transcendental extension of valued fields, where $K$ is equipped with a valuation $\nu$ of rank 1. That is, we assume given a rank 1 valuation $\nu$ of $K$ and its extension $\nu'$ to $L$. Let $(R_\nu,M_\nu,k_\nu)$ denote the valuation ring of $\nu$. The purpose of this paper is to present a refined version of MacLane's theory of key polynomials, similar to those considered by M. Vaqui\'e, and reminiscent of related objects studied by Abhyankar and Moh (approximate roots) and T.C. Kuo. Namely, we associate to $\iota$ a countable well ordered set $$ \mathbf{Q}=\{Q_i\}_{i\in\Lambda}\subset K[x]; $$ the $Q_i$ are called {\bf key polynomials}. Key polynomials $Q_i$ which have no immediate predecessor are called {\bf limit key polynomials}. Let $\beta_i=\nu'(Q_i)$. We give an explicit description of the limit key polynomials (which may be viewed as a generalization of the Artin--Schreier polynomials). We also give an upper bound on the order type of the set of key polynomials. Namely, we show that if $\operatorname{char}\ k_\nu=0$ then the set of key polynomials has order type at most $\omega$, while in the case $\operatorname{char}\ k_\nu=p>0$ this order type is bounded above by $\omega\times\omega$, where $\omega$ stands for the first infinite ordinal., Comment: arXiv admin note: substantial text overlap with arXiv:math/0605193

27 pages, 9 figures; We give a proof of a recent combinatorial conjecture due to the first author, which was discovered in the framework of commutative algebra. This result gives rise to new companions to the famous Andrews-Gordon identities. Our tools involve graded quotient rings, Durfee squares and rectangles for integer partitions, and $q$-series identities.

We analyse the representation of positive polynomials in terms of Sums of Squares. We provide a quantitative version of Putinar Positivstellensatz over a compact basic closed semialgebraic set S, with new polynomial bounds on the degree of the positivity certificates. These bounds involve a Łojasiewicz exponent associated to the description of S. We show that under Constraint Qualification Conditions, this Łojasiewicz exponent is equal to 1. We deduce new bounds on the convergence rate of the optima in Lasserre Sum-of-Squares hierarchy to the global optimum of a polynomial function on S and new bounds on the Hausdorff distance between the cone of truncated (probability) measures supported on S and the cone of truncated moment sequences, which are positive on the quadratic module of S.

International audience; This proceedings volume gathers together original articles and survey works that originate from presentations given at the conference Transient Transcendence in Transylvania, held in Brașov, Romania, from May 13th to 17th, 2019. The conference gathered international experts from various fields of mathematics and computer science, with diverse interests and viewpoints on transcendence. The covered topics are related to algebraic and transcendental aspects of special functions and special numbers arising in algebra, combinatorics, geometry and number theory. Besides contributions on key topics from invited speakers, this volume also brings selected papers from attendees.

A new Las Vegas algorithm is presented for the composition of two polynomials modulo a third one, over an arbitrary field. When the degrees of these polynomials are bounded by $n$, the algorithm uses $O(n^{1.43})$ field operations, breaking through the $3/2$ barrier in the exponent for the first time. The previous fastest algebraic algorithms, due to Brent and Kung in 1978, require $O(n^{1.63})$ field operations in general, and ${n^{3/2+o(1)}}$ field operations in the special case of power series over a field of large enough characteristic. If cubic-time matrix multiplication is used, the new algorithm runs in ${n^{5/3+o(1)}}$ operations, while previous ones run in $O(n^2)$ operations. Our approach relies on the computation of a matrix of algebraic relations that is typically of small size. Randomization is used to reduce arbitrary input to this favorable situation., 78 pages

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.

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., 28 pages

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; Additive polynomials play an important role in the local study of singularities in positive characteristic. For a base field k, the Hilbert-Samuel stratum of an affine cone over k is a linear space if k is perfect, but this is no longer true in general over non-perfect fields. H. Hironaka introduced additive subgroups in 1970 in order to quantify this phenomenon but little is known about classifying them except in small dimension.Building on joint works with A. Benito and A. Reguera on arc spaces, and with V. Cossart and B. Schober on the Hilbert-Samuel stratum, I will point out some of the questions arising for hypersurface cones defined by an additive polynomial of degree p in characteristic p>0.

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

We define logarithmic tangent sheaves associated with complete intersections in connection with Jacobian syzygies and distributions. We analyse the notions of local freeness, freeness and stability of these sheaves. We carry out a complete study of logarithmic sheaves associated with pencils of quadrics and compute their projective dimension from the classical invariants such as the Segre symbol and new invariants (splitting type and degree vector) designed for the classification of irregular pencils. This leads to a complete classification of free (equivalently, locally free) pencils of quadrics. Finally we produce examples of locally free, non free pencils of surfaces in P3 of any degree k at least 3, answering (in the negative) a question of Calvo-Andrade, Cerveau, Giraldo and Lins Neto about codimension foliations on P3 .

International audience; In this paper, we initiate the study of a discrete-time dynamical system modeling a trophic network connecting three types of plankton (phytoplankton, zooplankton, mixoplankton) and bacteria. The nonlinear operator V associated with this dynamical system is of type 4-Volterra quadratic stochastic operator with twelve parameters. We give conditions on the parameters under which this operator maps the five-dimensional standard simplex to itself and we find its fixed points. Moreover, we study the limit points of trajectories for this operator. For each situations we give some biological interpretations.

If $R$ is a commutative unital ring and $M$ is a unital $R$-module, then each element of $\operatorname{End}_R(M)$ determines a left $\operatorname{End}_{R}(M)[X]$-module structure on $\operatorname{End}_{R}(M)$, where $\operatorname{End}_{R}(M)$ is the $R$-algebra of endomorphisms of $M$ and $\operatorname{End}_{R}(M)[X] =\operatorname{End}_{R}(M)\otimes_RR[X]$. These structures provide a very short proof of the Cayley-Hamilton theorem, which may be viewed as a reformulation of the proof in Algebra by Serge Lang. Some generalisations of the Cayley-Hamilton theorem can be easily proved using the proposed method., v2: 4 pages. v1: 3 pages

This paper presents some 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.

A method for the construction of generalized integrity bases of invariantand covariant polynomials built from a set of three dimensional vectors under the SO(3)and O(3) symmetry is presented. The work is a follow–up to our previous article [G.Dhont and B. I. Zhilinskiı́ 2013. The action of the orthogonal group on planar vectors:invariants, covariants and syzygies. J. Phys. A: Math. Theor., 46, 455202] thatdealt with a set of two dimensional vectors under the action of the SO(2) and O(2)groups. The Molien generating functions for invariant rings and covariant modulesunder SO(3) are derived. Their expressions as one rational function are a useful guideto build integrity bases for rings of invariants and free modules of covariants. We showhow to deal with the Molien generating functions for non–free modules of covariantsand how to take advantage of their expressions as a sum of rational functions to obtaingeneralized integrity bases. The O(3) invariants and covariants bases are deducedfrom the SO(3) ones. The results can be used in quantum chemistry to describe thepotential energy or multipole moment hypersurfaces of molecules. In particular, thegeneralized integrity bases that are useful for the electric and magnetic quadrupolemoment hypersurfaces of tetratomic molecules are given for the first time.

In this paper we study the equations of the elimination ideal associated with $n+1$ generic multihomogeneous polynomials defined over a product of projective spaces of dimension $n$. We first prove a duality property and then make this duality explicit by introducing multigraded Sylvester forms. These results provide a partial generalization of similar properties that are known in the setting of homogeneous polynomial systems defined over a single projective space. As an important consequence, we derive a new family of elimination matrices that can be used for solving zero-dimensional multiprojective polynomial systems by means of linear algebra methods., To appear in Journal of Algebra