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1. Quelle est la place des exemples dans l'enseignement de l'algèbre abstraite ?

Author: Stalder, Julie
Subjects: [MATH.MATH-HO]Mathematics [math]/History and Overview [math.HO], [MATH.MATH-RA]Mathematics [math]/Rings and Algebras [math.RA]
Abstract: Teaching abstract algebra, seen as the study of structures and properties of structures, at a university level appears to be a challenge for both students and faculty. The professors in our study describe this passage through abstraction for the student body as a "killing game", an "impassable wall" or a "leap through abstraction". In this paper, based on a case study (Candy, 2020) we will investigate the choices of professors teaching abstract algebra at university. These professors were chosen because they teach abstract algebra at university in France, and they gave us access to their course corpus and agreed to be interviewed. In this article, we will choose to study in particular the teaching of the concept of ideal. An epistemological analysis will allow us to highlight its central role in the construction of abstract algebra. Then, using the anthropological theory of didactics (Chevallard, 1998), we will try to specify the place of examples and exercises in the praxis of the student body., Enseigner l'algèbre abstraite, vue comme l'étude des structures et des propriétés des structures, à un niveau universitaire apparaît être un challenge pour les corps étudiant et professoral. Les professeurs de notre étude décrivent ce passage à travers l'abstraction pour le corps étudiant comme un « jeu de massacre », un « mur infranchissable » ou encore « un saut à travers l'abstraction ». Dans cet article, en appui sur une étude de cas (Candy, 2020) nous allons étudier les choix de professeurs enseignant l'algèbre abstraite à l'université. Ces professeurs ont été choisis car ils enseignent l'algèbre abstraite à l'université en France, de plus ils nous ont donné accès à leur corpus de cours et ont accepté d'être interviewés. Dans cet article, nous choisirons d'étudier en particulier l'enseignement du concept d'idéal. Une analyse épistémologique nous permettra de mettre en évidence son rôle central dans la construction de l'algèbre abstraite. Puis, en utilisant la théorie anthropologique du didactique (Chevallard, 1998), nous chercherons à préciser la place des exemples et exercices dans la praxis du corps étudiant.
Published: 2023
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2. Sur les Initiaux et le Théorème Fondamental de la Géométrie Différentielle Tropicale Partielle

Author: Falkensteiner, Sebastian, Garay-Lopez, Cristhian Emmanuel, Haiech, Mercedes, Noordman, Marc Paul, Boulier, François, Toghani, Zeinab, Research Institute for Symbolic Computation (RISC), Johannes Kepler Universität Linz (JKU), Centro de Investigación en Matemáticas (CIMAT), Consejo Nacional de Ciencia y Tecnología [Mexico] (CONACYT), Institut de Recherche Mathématique de Rennes (IRMAR), Université de Rennes (UR)-Institut National des Sciences Appliquées - Rennes (INSA Rennes), Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-École normale supérieure - Rennes (ENS Rennes)-Université de Rennes 2 (UR2)-Centre National de la Recherche Scientifique (CNRS)-INSTITUT AGRO Agrocampus Ouest, Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro)-Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro), Bernoulli Institute for Mathematics and Computer Science and Artificial Intelligence, University of Groningen [Groningen], Centre de Recherche en Informatique, Signal et Automatique de Lille - UMR 9189 (CRIStAL), Centrale Lille-Université de Lille-Centre National de la Recherche Scientifique (CNRS), School of Mathematical Sciences [Queen Mary], University of London [London], ANR-17-CE40-0036,SYMBIONT,Méthodes symboliques pour les réseaux biologiques(2017), ANR-11-LABX-0020,LEBESGUE,Centre de Mathématiques Henri Lebesgue : fondements, interactions, applications et Formation(2011), Université de Rennes 1 (UR1), Université de Rennes (UNIV-RENNES)-Université de Rennes (UNIV-RENNES)-AGROCAMPUS OUEST, Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro)-Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro)-Institut National des Sciences Appliquées - Rennes (INSA Rennes), Institut National des Sciences Appliquées (INSA)-Université de Rennes (UNIV-RENNES)-Institut National des Sciences Appliquées (INSA)-École normale supérieure - Rennes (ENS Rennes)-Université de Rennes 2 (UR2), Université de Rennes (UNIV-RENNES)-Centre National de la Recherche Scientifique (CNRS), Université de Lille-Ecole Centrale de Lille-Centre National de la Recherche Scientifique (CNRS), AGROCAMPUS OUEST, Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro)-Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro)-Université de Rennes 1 (UR1), Université de Rennes (UNIV-RENNES)-Université de Rennes (UNIV-RENNES)-Université de Rennes 2 (UR2), Université de Rennes (UNIV-RENNES)-École normale supérieure - Rennes (ENS Rennes)-Centre National de la Recherche Scientifique (CNRS)-Institut National des Sciences Appliquées - Rennes (INSA Rennes), and Institut National des Sciences Appliquées (INSA)-Université de Rennes (UNIV-RENNES)-Institut National des Sciences Appliquées (INSA)
Subjects: [INFO.INFO-SC]Computer Science [cs]/Symbolic Computation [cs.SC], [MATH.MATH-AC]Mathematics [math]/Commutative Algebra [math.AC], [MATH.MATH-RA]Mathematics [math]/Rings and Algebras [math.RA], ComputingMilieux_MISCELLANEOUS
Abstract: International audience
Published: 2023
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3. Algebraic Number Theory with Elementary Galois Theory

Author: Coleman, Rodney, Zwald, Laurent, Calcul Algébrique et Symbolique, Sécurité, Systèmes Complexes, Codes et Cryptologie (CASC), Laboratoire Jean Kuntzmann (LJK), Institut National de Recherche en Informatique et en Automatique (Inria)-Centre National de la Recherche Scientifique (CNRS)-Université Grenoble Alpes (UGA)-Institut polytechnique de Grenoble - Grenoble Institute of Technology (Grenoble INP ), Université Grenoble Alpes (UGA)-Institut National de Recherche en Informatique et en Automatique (Inria)-Centre National de la Recherche Scientifique (CNRS)-Université Grenoble Alpes (UGA)-Institut polytechnique de Grenoble - Grenoble Institute of Technology (Grenoble INP ), Université Grenoble Alpes (UGA), and Statistique pour le Vivant et l’Homme (SVH)
Subjects: [MATH.MATH-RA]Mathematics [math]/Rings and Algebras [math.RA], Algebraic Number Theory, Galois Theory, [MATH.MATH-NT]Mathematics [math]/Number Theory [math.NT]
Abstract: International audience
Published: 2023

4. Faithful flatness of Hopf algebras over coideal subalgebras with a conditional expectation

Author: Bichon, Julien, Laboratoire de Mathématiques Blaise Pascal (LMBP), and Centre National de la Recherche Scientifique (CNRS)-Université Clermont Auvergne (UCA)
Subjects: 16T05, Rings and Algebras (math.RA), [MATH.MATH-RA]Mathematics [math]/Rings and Algebras [math.RA], Mathematics - Quantum Algebra, FOS: Mathematics, [MATH.MATH-QA]Mathematics [math]/Quantum Algebra [math.QA], Quantum Algebra (math.QA), Mathematics - Rings and Algebras
Abstract: Let $H$ be a Hopf algebra and let $A\subset H$ be a right coideal subalgebra. We show that if $A$ is a direct summand in $H$ as a right $A$-module, then $H$ is faithfully flat as a right $A$-module.
Published: 2023

5. A Gröbner basis approach to m-dimensional skew (consta-) cyclic codes: Dedicated to André Leroy on his retirement

Author: Geiselmann, Willi, Ulmer, Félix, Institut für Theoretische Informatik (ITI), Karlsruher Institut für Technologie (KIT), Institut de Recherche Mathématique de Rennes (IRMAR), Université de Rennes (UR)-Institut National des Sciences Appliquées - Rennes (INSA Rennes), Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-École normale supérieure - Rennes (ENS Rennes)-Université de Rennes 2 (UR2)-Centre National de la Recherche Scientifique (CNRS)-Institut Agro Rennes Angers, and Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro)-Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro)
Subjects: [INFO.INFO-SC]Computer Science [cs]/Symbolic Computation [cs.SC], [INFO.INFO-IT]Computer Science [cs]/Information Theory [cs.IT], [MATH.MATH-RA]Mathematics [math]/Rings and Algebras [math.RA]
Abstract: Algebraic linear codes J/I ⊂ Fq[X1,. .. Xm]/I that are finite dimension quotients of multivariate polynomial rings, have been studied intensively; in particular m-dimensional cyclic codes where the ideal I is (X n 1 1 − 1,. .. , X nm m − 1). Recently this notion has been extended to two dimensional skew cyclic codes using multivariate skew polynomial rings over fields [12, 16, 18] and even over rings [15]. In this paper we use a Gröbner basis approach in order to generalize algebraic linear codes and m-dimensional cyclic codes to the skew polynomial rings setting. The approach encompass all previous results on m-dimensional (consta-)cyclic codes and allows for many generalizations.
Published: 2022

6. ON THE DOUBLE OF THE JORDAN PLANE

Author: Nicolás Andruskiewitsch, François Dumas, Héctor Martín Peña Pollastri, Laboratoire de Mathématiques Blaise Pascal (LMBP), and Université Clermont Auvergne [2017-2020] (UCA [2017-2020])-Centre National de la Recherche Scientifique (CNRS)
Subjects: 16T05, Nichols algebra, Rings and Algebras (math.RA), General Mathematics, Drinfeld double, Jordan plane, [MATH.MATH-RA]Mathematics [math]/Rings and Algebras [math.RA], FOS: Mathematics, [MATH.MATH-QA]Mathematics [math]/Quantum Algebra [math.QA], Mathematics - Rings and Algebras, Weyl algebra
Abstract: International audience; We compute the simple finite-dimensional modules and the center of the Drinfeld double of the Jordan plane introduced in [AP1] assuming that the characteristic is zero.
Published: 2022

7. Étude d'une propriété semblable à la divisibilité

Author: Khanfir, Robin, Seguin, Béranger, Laboratoire de Probabilités, Statistique et Modélisation (LPSM (UMR_8001)), Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS)-Université Paris Cité (UPCité), Sorbonne Université (SU), Département de Mathématiques et Applications - ENS Paris (DMA), École normale supérieure - Paris (ENS-PSL), Université Paris sciences et lettres (PSL)-Université Paris sciences et lettres (PSL)-Centre National de la Recherche Scientifique (CNRS), Laboratoire Paul Painlevé (LPP), Université de Lille-Centre National de la Recherche Scientifique (CNRS), and Université de Lille
Subjects: Noncommutative rings, Noncommutative rings Simple algebras Ore condition Ore extension Mathematics Subject Classification (MSC 2010): 16U20 13N10 12E15 16P40, 16P40, V-domain, [MATH.MATH-RA]Mathematics [math]/Rings and Algebras [math.RA], 12E15, Mathematics - Rings and Algebras, 16U20, 13N10, 12E15, 16P40, Ore condition, Rings and Algebras (math.RA), Ore extensions, Simple algebras, FOS: Mathematics, 13N10, Ore extension Mathematics Subject Classification (MSC 2010): 16U20, MSC 16U20, 13N10, 12E15, 16P40
Abstract: We introduce a weak division-like property for noncommutative rings: a nontrivial ring is fadelian if for all nonzero $a,x$ there exist $b,c$ such that $x=ab+ca$. We prove properties of fadelian rings, and construct examples of such rings which are not division rings, as well as non-Noetherian and non-Ore examples. We have also formalized some of these results in the Lean proof assistant., 15 pages. 1 figure, 1 appendix (Lean code). Comments welcome!
Published: 2022

8. Factorization of skew polynomials over k((u))

Author: Le Borgne, Jérémy, Université de Rennes (UR), Institut de Recherche Mathématique de Rennes (IRMAR), Université de Rennes (UR)-Institut National des Sciences Appliquées - Rennes (INSA Rennes), Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-École normale supérieure - Rennes (ENS Rennes)-Université de Rennes 2 (UR2)-Centre National de la Recherche Scientifique (CNRS)-Institut Agro Rennes Angers, Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro)-Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro), and École normale supérieure - Rennes (ENS Rennes)
Subjects: Skew polynomial rings, [MATH.MATH-RT]Mathematics [math]/Representation Theory [math.RT], Rings and Algebras (math.RA), Galois representations, [MATH.MATH-AC]Mathematics [math]/Commutative Algebra [math.AC], [MATH.MATH-RA]Mathematics [math]/Rings and Algebras [math.RA], FOS: Mathematics, Newton polygons, Mathematics - Rings and Algebras, Representation Theory (math.RT), Commutative Algebra (math.AC), Mathematics - Commutative Algebra, Mathematics - Representation Theory
Abstract: 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.
Published: 2022

9. Une approche par opérateurs intégro-diférentiels des systèmes différentiels linéaires

Author: A. Quadrat, OUtils de Résolution Algébriques pour la Géométrie et ses ApplicatioNs (OURAGAN), Inria de Paris, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria), Institut de Mathématiques de Jussieu - Paris Rive Gauche (IMJ-PRG (UMR_7586)), and Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS)-Université Paris Cité (UPCité)
Subjects: [INFO.INFO-SC]Computer Science [cs]/Symbolic Computation [cs.SC], Parametrization, ACM: I.: Computing Methodologies/I.1: SYMBOLIC AND ALGEBRAIC MANIPULATION/I.1.2: Algorithms/I.1.2.0: Algebraic algorithms, [MATH.MATH-RA]Mathematics [math]/Rings and Algebras [math.RA], [MATH.MATH-OA]Mathematics [math]/Operator Algebras [math.OA], Linear systems, Behaviour theory, Reachability, Rings of integro-differential operators, [SPI.AUTO]Engineering Sciences [physics]/Automatic, Algebraic analysis, ACM: G.: Mathematics of Computing/G.1: NUMERICAL ANALYSIS/G.1.9: Integral Equations/G.1.9.2: Integro-differential equations, Control and Systems Engineering, Polynomial methods, ACM: I.: Computing Methodologies/I.1: SYMBOLIC AND ALGEBRAIC MANIPULATION/I.1.1: Expressions and Their Representation
Abstract: International audience; In this paper, we initiate a new algebraic analysis approach to linear differential systems based on rings of integro-differential operators. Within this algebraic analysis approach, we first interpret the method of variations of constants as an operator identity. Using this result, we show that the module associated with a state-space representation of a linear system is the same as the one associated with its standard convolution representation. This finitely presented module over the ring of integro-differential operators is proved to be stably free. Finally, we show how the reachability property can be expressed within this algebraic analysis approach.
Published: 2022
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10. IC rings and transitivity of perspectivity

Author: Pace P. Nielsen, Xavier Mary, Modélisation aléatoire de Paris X (MODAL'X), Université Paris Nanterre (UPN), Fédération Parisienne de Modélisation Mathématique (FP2M), and Centre National de la Recherche Scientifique (CNRS)
Subjects: Transitive relation, Pure mathematics, Ring (mathematics), Algebra and Number Theory, Perspective (geometry), Applied Mathematics, Perspectivity, [MATH.MATH-RA]Mathematics [math]/Rings and Algebras [math.RA], Construct (philosophy), [MATH.MATH-GR]Mathematics [math]/Group Theory [math.GR], Mathematics
Abstract: International audience; We construct an example of an IC ring where perspectivity is transitive, but not all isomorphic idempotents are perspective. We also develop new criteria for checking perspectivity of idempotents in rings.
Published: 2022
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11. Semiring systems arising from hyperrings

Author: Akian, Marianne, Gaubert, Stephane, Rowen, Louis, TROPICAL (TROPICAL), Centre de Mathématiques Appliquées - Ecole Polytechnique (CMAP), École polytechnique (X)-Centre National de la Recherche Scientifique (CNRS)-École polytechnique (X)-Centre National de la Recherche Scientifique (CNRS)-Inria Saclay - Ile de France, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria), Department of Mathematics [Bar-Ilan], and Bar-Ilan University [Israël]
Subjects: symmetrization, bipotent, triple, [MATH.MATH-RA]Mathematics [math]/Rings and Algebras [math.RA], geometric triple, tropical, semifield, Mathematics - Rings and Algebras, system, hyperfield, semiring, Primary 06F20, 14T10, 15A80, 16Y20, 16Y60, Secondary 20N20, 15A24, surpassing relation, fuzzy ring, tangible balancing, negation map, Rings and Algebras (math.RA), semigroup, module, FOS: Mathematics, metatangible, height
Abstract: Hyperfields and systems are two algebraic frameworks which have been developed to provide a unified approach to classical and tropical structures. All hyperfields, and more generally hyperrings, can be represented by systems. We show that, conversely, we show that the systems arising in this way, called hypersystems, are characterized by certain elimination axioms. Systems are preserved under standard algebraic constructions; for instance matrices and polynomials over hypersystems are systems, but not hypersystems. We illustrate these results by discussing several examples of systems and hyperfields, and constructions like matroids over systems.
Published: 2022

12. Bicomplex numbers as a normal complexified f-algebra

Author: Gargoubi, Hichem, Kossentini, Sayed, Université de Tunis, Université de Tunis - El Manar II, Université de Tunis El Manar (UTM), and GAR, Hich
Subjects: Bicomplex gamma function, Mathematics::Complex Variables, General Mathematics, [MATH.MATH-RA]Mathematics [math]/Rings and Algebras [math.RA], Riesz space, [MATH] Mathematics [math], Lattices, Mathematics - Rings and Algebras, Bicomplex numbers, Functional Analysis (math.FA), Mathematics - Functional Analysis, Nonlinear Sciences::Exactly Solvable and Integrable Systems, Rings and Algebras (math.RA), Mathematics::K-Theory and Homology, [MATH.MATH-RA] Mathematics [math]/Rings and Algebras [math.RA], FOS: Mathematics, f -algebra, Hyperbolic numbers, [MATH]Mathematics [math], Bicomplex zeta function
Abstract: The algebra B of bicomplex numbers is viewed as a complexification of the Archimedean f-algebra of hyperbolic numbers D. This lattice-theoretic approach allows us to establish new properties of the so-called D-norms. In particular, we show that D-norms generate the same topology in B. We develop the D-trigonometric form of a bicomplex number which leads us to a geometric interpretation of the nth roots of a bicomplex number in terms of polyhedral tori. We use the concepts developed, in particular that of Riesz subnorm of a D-norm, to study the uniform convergence of the bicomplex zeta and gamma functions. The main result of this paper is the generalization to the bicomplex case of the Riemann functional equation and Euler's reflection formula., 23 pages, 1 figure. Submitted
Published: 2022
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13. On the Ore extension ring of differential time-varying delay operators

Author: Quadrat, Alban, Ushirobira, Rosane, OUtils de Résolution Algébriques pour la Géométrie et ses ApplicatioNs [OURAGAN], Institut de Mathématiques de Jussieu - Paris Rive Gauche [IMJ-PRG (UMR_7586)], Finite-time control and estimation for distributed systems [VALSE], OUtils de Résolution Algébriques pour la Géométrie et ses ApplicatioNs (OURAGAN), Inria de Paris, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria), Institut de Mathématiques de Jussieu - Paris Rive Gauche (IMJ-PRG (UMR_7586)), Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS)-Université Paris Cité (UPCité), Finite-time control and estimation for distributed systems (VALSE), Inria Lille - Nord Europe, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-Centre de Recherche en Informatique, Signal et Automatique de Lille - UMR 9189 (CRIStAL), and Centrale Lille-Université de Lille-Centre National de la Recherche Scientifique (CNRS)-Centrale Lille-Université de Lille-Centre National de la Recherche Scientifique (CNRS)
Subjects: ACM: G.: Mathematics of Computing/G.1: NUMERICAL ANALYSIS/G.1.7: Ordinary Differential Equations, [INFO.INFO-SC]Computer Science [cs]/Symbolic Computation [cs.SC], ACM: G.: Mathematics of Computing/G.1: NUMERICAL ANALYSIS/G.1.9: Integral Equations/G.1.9.0: Delay equations, ACM: I.: Computing Methodologies/I.1: SYMBOLIC AND ALGEBRAIC MANIPULATION/I.1.2: Algorithms/I.1.2.0: Algebraic algorithms, [MATH.MATH-RA]Mathematics [math]/Rings and Algebras [math.RA], [MATH.MATH-OA]Mathematics [math]/Operator Algebras [math.OA], [MATH.MATH-OC]Mathematics [math]/Optimization and Control [math.OC], ACM: I.: Computing Methodologies/I.1: SYMBOLIC AND ALGEBRAIC MANIPULATION/I.1.1: Expressions and Their Representation, [SPI.AUTO]Engineering Sciences [physics]/Automatic
Abstract: International audience; In this work, we propose an algebraic method to study linear differential time-varying delay (DTVD) systems. Our goal is to give an effective construction of the ring of DTVD operators as an Ore extension, thanks to the concept of skew polynomial rings developed by Ore in the 30s. Some algebraic properties of the DTVD operators ring are analyzed, such as its Noetherianity, its homological and Krull dimensions, and the existence of Gröbner bases, all given in terms of the time-varying delay function. The algebraic analysis framework for linear systems theory allows us to study linear DTVD systems and essential properties such as the existence of autonomous elements, controllability, parametrizability, flatness, etc., through methods coming from module theory, homological algebra, and constructive algebra.
Published: 2022
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14. Calcul de la norme L∞ des systèmes linéaires de dimension finie

Author: Grace Younes, Institut de Mathématiques de Jussieu - Paris Rive Gauche (IMJ-PRG (UMR_7586)), Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS)-Université Paris Cité (UPCité), OUtils de Résolution Algébriques pour la Géométrie et ses ApplicatioNs (OURAGAN), Inria de Paris, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria), Sorbonne Université, Fabrice Rouillier, and Alban Quadrat
Subjects: [INFO.INFO-SC]Computer Science [cs]/Symbolic Computation [cs.SC], Théorie du contrôle, L∞ norm computation, [MATH.MATH-RA]Mathematics [math]/Rings and Algebras [math.RA], Polynomial systems, Solutions réelles maximales, Calcul de complexité, Control theory, Calcul symbolique, Maximal real roots, Implementation, Systèmes polynomiaux, Calcul de la norme L∞, [INFO.INFO-RB]Computer Science [cs]/Robotics [cs.RO], Complexity computation, Implémentation, Symbolicla computation
Abstract: In this dissertation, we study the computation of the L∞-norm of finite-dimensional linear time-invariant systems. This problem is first reduced to the computation of the maximal y-projection of the real solutions (x, y) of a bivariate polynomial equations system Σ = {P = 0, ∂P/∂x = 0}, where P ∈ Z[x, y]. Then, we use standard computer algebra methods to solve this problem. In particular, we alternatively study a method based on rational univariate representations, a method based on root separation, and finally a method first based on the sign variation of the leading coefficients of a signed subresultant sequence (Sturm-Habicht) and on the identification of an isolating interval for the maximal y-projection of the real solutions of Σ. We then compute the worst-case bit complexity of each method and compare their theoretical behavior. We also implement each method in Maple and compare their practical behavior (average complexity). A generalization of the above algorithms is finally proposed to the case where the polynomial P also depends on a set of parameters α = [α1, . . . , αd] ∈ Rd. To do that, we solve the problem using the notion of the Cylindrical Algebraic Decomposition, well-known in algebraic geometry.; Dans cette thèse, nous étudions le problème du calcul de la norme L∞ des systèmes de dimension finie, linéaires et invariants dans le temps. Ce problème est ramené au calcul de la y-projection maximale des solutions réelles (x, y) d’un système d’équations polynomiales bivariées Σ = {P = 0, ∂P/∂x = 0}, où P ∈ Z[x, y]. Nous utilisons alors des méthodes classiques de calcul formel pour résoudre ce problème. En particulier, nous étudions alternativement une méthode basée sur des représentations univariées rationnelles, une méthode basée sur la séparation des racines, et enfin une méthode basée sur la variation de signes des coefficients dominants d’une suite signée de sous-résultants (suite de Sturm-Habicht) et l’identification d’un intervalle isolant pour la y-projection maximale des solutions réelles de Σ. Nous calculons ensuite la complexité binaire dans le pire des cas de chacune des méthodes proposées et nous comparons leur comportement théorique. Enfin, nous implémentons chacune des méthodes sous Maple et nous comparons leur comportement pratique (complexité moyenne). Une généralisation des algorithmes précédents au cas de polynômes P dépendants aussi de paramètres α = [α1, . . . , αd] ∈ Rd est finalement proposée. Pour cela, nous résolvons le problème en utilisant la notion de Décomposition Cylindrique Algébrique, classique en géométrie algébrique.
Published: 2022

15. Special clean elements, perspective elements and perspective rings

Author: Xavier Mary, Modélisation aléatoire de Paris X (MODAL'X), Université Paris Nanterre (UPN), Fédération Parisienne de Modélisation Mathématique (FP2M), and Centre National de la Recherche Scientifique (CNRS)
Subjects: Algebra and Number Theory, Applied Mathematics, [MATH.MATH-RA]Mathematics [math]/Rings and Algebras [math.RA], [MATH.MATH-GR]Mathematics [math]/Group Theory [math.GR]
Abstract: International audience; Motivated by the idea of perspectivity of rings and modules, we introduce perspective elements of a ring. We notably prove that perspective elements form a proper subset of special clean elements, and that perspectivity of elements is a left-right symmetric property. An equational characterization of perspective elements is also given and examples are provided.
Published: 2022
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16. Reduced typed angularly decorated planar rooted trees and generalized tridendriform algebras

Author: Foissy, Loïc, Peng, Xiao-Song, Laboratoire de Mathématiques Pures et Appliquées Joseph Liouville (LMPA), Université du Littoral Côte d'Opale (ULCO), Department of Mathematics and Statistics [Lanzhou], and Lanzhou University
Subjects: [MATH.MATH-RA]Mathematics [math]/Rings and Algebras [math.RA], Mathematics - Rings and Algebras
Abstract: We introduce a generalization of tridendriform algebras, where each of the three products are replaced by a family of products indexed by a set $\Omega$. We study the needed structure on $\Omega$ for free $\Omega$-tridendriform algebras to be built on Schr{\"o}der trees (as it is the case in the classical case), with convenient decorations on their leaves. We obtain in this way extended triassociative semigroups. We describe commutative $\Omega$-tridendriform algebras in terms of typed words. We also study links with generalizations of Rota-Baxter algebras and describe the Koszul duals of the corresponding operads.
Published: 2021

17. Typed angularly decorated planar rooted trees and generalized Rota-Baxter algebras

Author: Foissy, Loïc, Peng, Xiao-Song, Laboratoire de Mathématiques Pures et Appliquées Joseph Liouville (LMPA), Université du Littoral Côte d'Opale (ULCO), Department of Mathematics and Statistics [Lanzhou], and Lanzhou University
Subjects: November 23, 08B20, 2021. 1991 Mathematics Subject Classification. 17B38, [MATH.MATH-RA]Mathematics [math]/Rings and Algebras [math.RA], Mathematics::History and Overview, Mathematics::Rings and Algebras, Mathematics - Rings and Algebras, 05C05, Nonlinear Sciences::Exactly Solvable and Integrable Systems, Rings and Algebras (math.RA), Mathematics::Category Theory, Mathematics::Quantum Algebra, FOS: Mathematics, 16S10
Abstract: We introduce a generalization of parametrized Rota-Baxter algebras, which includes family and matching Rota-Baxter algebras. We study the structure needed on the set $\Omega$ of parameters in order to obtain that free Rota-Baxter algebras are described in terms of typed and angularly decorated planar rooted trees: we obtain the notion of $\lambda$-extended diassociative semigroup, which includes sets (for matching Rota-Baxter algebras) and semigroups (for family Rota-Baxter algebras), and many other examples. We also describe free commutative $\Omega$-Rota-Baxter algebras generated by a commutative algebra A in terms of typed words.
Published: 2021

18. Growth of structure constants of free Lie algebras relative to Hall bases

Author: Karine Beauchard, Jérémy Le Borgne, Frédéric Marbach, Centre National de la Recherche Scientifique (CNRS), Institut de Recherche Mathématique de Rennes (IRMAR), AGROCAMPUS OUEST, Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro)-Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro)-Université de Rennes 1 (UR1), Université de Rennes (UNIV-RENNES)-Université de Rennes (UNIV-RENNES)-Université de Rennes 2 (UR2), Université de Rennes (UNIV-RENNES)-École normale supérieure - Rennes (ENS Rennes)-Centre National de la Recherche Scientifique (CNRS)-Institut National des Sciences Appliquées - Rennes (INSA Rennes), Institut National des Sciences Appliquées (INSA)-Université de Rennes (UNIV-RENNES)-Institut National des Sciences Appliquées (INSA), Institut National des Sciences Appliquées - Rennes (INSA Rennes), Institut National des Sciences Appliquées (INSA)-Université de Rennes (UNIV-RENNES)-Institut National des Sciences Appliquées (INSA)-Université de Rennes (UNIV-RENNES)-Centre National de la Recherche Scientifique (CNRS)-École normale supérieure - Rennes (ENS Rennes)-Université de Rennes 2 (UR2), Université de Rennes (UNIV-RENNES)-Université de Rennes 1 (UR1), Université de Rennes (UNIV-RENNES)-AGROCAMPUS OUEST, Université de Rennes (UR)-Institut National des Sciences Appliquées - Rennes (INSA Rennes), Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-École normale supérieure - Rennes (ENS Rennes)-Université de Rennes 2 (UR2)-Centre National de la Recherche Scientifique (CNRS)-INSTITUT AGRO Agrocampus Ouest, Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro)-Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro), ANR-11-LABX-0020,LEBESGUE,Centre de Mathématiques Henri Lebesgue : fondements, interactions, applications et Formation(2011), Université de Rennes 1 (UR1), and Institut National des Sciences Appliquées (INSA)-Université de Rennes (UNIV-RENNES)-Institut National des Sciences Appliquées (INSA)-INSTITUT AGRO Agrocampus Ouest
Subjects: Algebra and Number Theory, Rings and Algebras (math.RA), [MATH.MATH-RA]Mathematics [math]/Rings and Algebras [math.RA], [MATH.MATH-CO]Mathematics [math]/Combinatorics [math.CO], FOS: Mathematics, Mathematics - Combinatorics, Combinatorics (math.CO), Mathematics - Rings and Algebras
Abstract: We derive a priori bounds on the size of the structure constants of the free Lie algebra over a set of indeterminates, relative to its Hall bases. We investigate their asymptotic growth, especially as a function of the length of the involved Lie brackets. First, using the classical recursive decomposition algorithm, we obtain a rough upper bound valid for all Hall bases. We then introduce new notions (which we call alphabetic subsets and relative foldings) related to structural properties of the Lie brackets created by the algorithm, which allow us to prove a sharp upper bound for the general case. We also prove that the length of the relative folding provides a strictly decreasing indexation of the recursive rewriting algorithm. Moreover, we derive lower bounds on the structure constants proving that they grow at least geometrically in all Hall bases. Second, for the celebrated historical length-compatible Hall bases and the Lyndon basis, we prove tighter sharp upper bounds, which turn out to be geometric in the length of the brackets. Third, we construct two new Hall bases, illustrating two opposite behaviors in the two-indeterminates case. One is designed so that its structure constants have the minimal growth possible, matching exactly the general lower bound, linked with the Fibonacci sequence. The other one is designed so that its structure constants grow super-geometrically. Eventually, we investigate asymmetric growth bounds which isolate the role of one particular indeterminate. Despite the existence of super-geometric Hall bases, we prove that the asymmetric growth with respect to each fixed indeterminate is uniformly at most geometric in all Hall bases., Fixed typos
Published: 2021
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19. Duals of linearized Reed-Solomon codes

Author: Caruso, Xavier, Durand, Amaury, Institut de Mathématiques de Bordeaux (IMB), Université Bordeaux Segalen - Bordeaux 2-Université Sciences et Technologies - Bordeaux 1-Université de Bordeaux (UB)-Institut Polytechnique de Bordeaux (Bordeaux INP)-Centre National de la Recherche Scientifique (CNRS), Lithe and fast algorithmic number theory (LFANT), Université Bordeaux Segalen - Bordeaux 2-Université Sciences et Technologies - Bordeaux 1-Université de Bordeaux (UB)-Institut Polytechnique de Bordeaux (Bordeaux INP)-Centre National de la Recherche Scientifique (CNRS)-Université Bordeaux Segalen - Bordeaux 2-Université Sciences et Technologies - Bordeaux 1-Université de Bordeaux (UB)-Institut Polytechnique de Bordeaux (Bordeaux INP)-Centre National de la Recherche Scientifique (CNRS)-Inria Bordeaux - Sud-Ouest, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria), Université Bordeaux Segalen - Bordeaux 2-Université Sciences et Technologies - Bordeaux 1 (UB)-Université de Bordeaux (UB)-Institut Polytechnique de Bordeaux (Bordeaux INP)-Centre National de la Recherche Scientifique (CNRS)-Université Bordeaux Segalen - Bordeaux 2-Université Sciences et Technologies - Bordeaux 1 (UB)-Université de Bordeaux (UB)-Institut Polytechnique de Bordeaux (Bordeaux INP)-Centre National de la Recherche Scientifique (CNRS)-Inria Bordeaux - Sud-Ouest, Université Bordeaux Segalen - Bordeaux 2-Université Sciences et Technologies - Bordeaux 1 (UB)-Université de Bordeaux (UB)-Institut Polytechnique de Bordeaux (Bordeaux INP)-Centre National de la Recherche Scientifique (CNRS), and ANR-18-CE40-0026,CLap-CLap,Correspondance de Langlands p-adique : une approche constructive et algorithmique(2018)
Subjects: FOS: Computer and information sciences, Information Theory (cs.IT), Applied Mathematics, Computer Science - Information Theory, [MATH.MATH-RA]Mathematics [math]/Rings and Algebras [math.RA], Data_CODINGANDINFORMATIONTHEORY, Mathematics - Rings and Algebras, Computer Science Applications, Rings and Algebras (math.RA), [INFO.INFO-IT]Computer Science [cs]/Information Theory [cs.IT], ComputingMethodologies_SYMBOLICANDALGEBRAICMANIPULATION, FOS: Mathematics, Hardware_ARITHMETICANDLOGICSTRUCTURES, Computer Science::Information Theory
Abstract: International audience; We give a description of the duals of linearized Reed-Solomon codes in terms of codes obtained by taking residues of Ore rational functions. Our construction shows in particular that, under some assumptions on the base field, the class of linearized Reed-Solomon codes is stable under duality. As a byproduct of our work, we develop a theory of residues in the Ore setting.
Published: 2021
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20. Generalised Howe dualityand injectivity of induction: the symplectic case

Author: Gerber, Thomas, Guilhot, Jeremie, Lecouvey, Cédric, Ecole Polytechnique Fédérale de Lausanne (EPFL), Institut Denis Poisson (IDP), Centre National de la Recherche Scientifique (CNRS)-Université de Tours (UT)-Université d'Orléans (UO), and Centre National de la Recherche Scientifique (CNRS)-Université de Tours-Université d'Orléans (UO)
Subjects: [MATH.MATH-RT]Mathematics [math]/Representation Theory [math.RT], Rings and Algebras (math.RA), [MATH.MATH-RA]Mathematics [math]/Rings and Algebras [math.RA], [MATH.MATH-CO]Mathematics [math]/Combinatorics [math.CO], FOS: Mathematics, Mathematics - Combinatorics, Combinatorics (math.CO), Mathematics - Rings and Algebras, Representation Theory (math.RT), Mathematics - Representation Theory
Abstract: We study the symplectic Howe duality using two new and independent combinatorial methods: via determinantal formulae on the one hand, and via (bi)crystals on the other hand. The first approach allows us to establish a generalised version where weight multiplicities are replaced by branching coefficients. In turn, this generalised Howe duality is used to prove the injectivity of induction for Levi branchings as previously conjectured by the last two authors.
Published: 2021

21. R-EQUIVALENCE ON GROUP SCHEMES AND NON STABLE K 1 -FUNCTORS

Author: Gille, Philippe, Stavrova, A, Institut Camille Jordan [Villeurbanne] (ICJ), École Centrale de Lyon (ECL), Université de Lyon-Université de Lyon-Université Claude Bernard Lyon 1 (UCBL), Université de Lyon-Institut National des Sciences Appliquées de Lyon (INSA Lyon), Université de Lyon-Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-Université Jean Monnet [Saint-Étienne] (UJM)-Centre National de la Recherche Scientifique (CNRS), Chebyshev Laboratory, St. Petersburg State University, Université de Lyon-Université Jean Monnet [Saint-Étienne] (UJM)-Institut National des Sciences Appliquées de Lyon (INSA Lyon), and Université de Lyon-Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-Centre National de la Recherche Scientifique (CNRS)
Subjects: resolution property. MSC 2000: 14L15, Group schemes, tori, [MATH.MATH-RA]Mathematics [math]/Rings and Algebras [math.RA], representations, 20G35, R-equivalence, [MATH.MATH-AG]Mathematics [math]/Algebraic Geometry [math.AG], Reductive group schemes, Whitehead groups and K 1, Karoubi-Villamayor Ktheory
Abstract: We define R-equivalence for group schemes over a semilocal ring and relate this with rational properties. Two main cases are investigated: tori and isotropic semisimple simply connected group schemes where we show in certain cases that R-equivalence coincide with Karoubi-Villamayor equivalence and is also related to the Kneser-Tits problem in this setting. Finally we construct specialization maps for R-equivalence in the case of regular algebras containing a field.
Published: 2021

22. Zinbiel algebras and multiple zeta values

Author: Frédéric Chapoton, Institut de Recherche Mathématique Avancée (IRMA), Université de Strasbourg (UNISTRA)-Centre National de la Recherche Scientifique (CNRS), and ANR-19-CE48-0011,COMBINE,Combinatoire enumerative en interaction avec l'algebre, la theorie des nombres et la physique(2019)
Subjects: Mathematics - Number Theory, General Mathematics, Mathematics::Rings and Algebras, [MATH.MATH-RA]Mathematics [math]/Rings and Algebras [math.RA], zinbiel algebra, MSC2020 17A50, 11M32, 68R15, Mathematics - Rings and Algebras, multiple zeta values, [MATH.MATH-NT]Mathematics [math]/Number Theory [math.NT], shuffle algebra
Abstract: We build, using the notion of zinbiel algebra, some commutative subalgebras $C_{u,v}$ inside an algebra of formal iterated integrals. There is a quotient map from this algebra of formal iterated integrals to the algebra of motivic multiple zeta values. Restricting this quotient map to the subalgebras $C_{u,v}$ of A$_{1,0}$ gives a morphism of graded commutative algebras with the same graded dimension. This is conjectured to be generically an isomorphism. When u+v = 0, the image is instead a sub-algebra of the algebra of motivic multiple zeta values., Comment: 11 pages, 1 figure
Published: 2021

23. Post-Lie-Magnus expansion and BCH-recursion

Author: Al-Kaabi, Mahdi Jasim Hasan, Ebrahimi-Fard, Kurusch, Manchon, Dominique, Laboratoire de Mathématiques Blaise Pascal (LMBP), Centre National de la Recherche Scientifique (CNRS)-Université Clermont Auvergne (UCA), Al-Mustansiriyah University, and Norwegian University of Science and Technology (NTNU)
Subjects: 16T05, 16T10, 16T30, 17A30, rooted trees. MSC Classification: 16T05, Mathematics::Rings and Algebras, Mathematics::History and Overview, [MATH.MATH-RA]Mathematics [math]/Rings and Algebras [math.RA], Mathematics - Rings and Algebras, Magnus expansion, BCH-formula, Nonlinear Sciences::Exactly Solvable and Integrable Systems, Rota-Baxter algebra, Rings and Algebras (math.RA), Condensed Matter::Superconductivity, Mathematics::Quantum Algebra, Rota--Baxter algebra, FOS: Mathematics, pre-Lie algebra, Post-Lie algebra, rooted trees
Abstract: International audience; We identify the Baker-Campbell-Hausdorff recursion driven by a weight$\lambda=1$ Rota-Baxter operator with the Magnus expansion relativeto the post-Lie structure naturally associated to the correspondingRota-Baxter algebra. Post-Lie Magnus expansion and BCH-recursionare reviewed before the proof of the main result.
Published: 2021

24. Mathematical morphology meets Deep Learning

Author: Velasco-Forero, Santiago, Blusseau, Samy, Sangalli, Mateus, Centre de Morphologie Mathématique (CMM), MINES ParisTech - École nationale supérieure des mines de Paris, Université Paris sciences et lettres (PSL)-Université Paris sciences et lettres (PSL), and Université Paris sciences et lettres (PSL)
Subjects: [INFO.INFO-TS]Computer Science [cs]/Signal and Image Processing, [MATH.MATH-ST]Mathematics [math]/Statistics [math.ST], [MATH.MATH-RA]Mathematics [math]/Rings and Algebras [math.RA], [INFO.INFO-NE]Computer Science [cs]/Neural and Evolutionary Computing [cs.NE], ComputingMilieux_MISCELLANEOUS, [INFO.INFO-AI]Computer Science [cs]/Artificial Intelligence [cs.AI]
Abstract: International audience
Published: 2021

25. SPRINGER'S ODD DEGREE EXTENSION THEOREM FOR QUADRATIC FORMS OVER SEMILOCAL RINGS

Author: Gille, Philippe, Neher, Erhard, Institut Camille Jordan [Villeurbanne] (ICJ), École Centrale de Lyon (ECL), Université de Lyon-Université de Lyon-Université Claude Bernard Lyon 1 (UCBL), Université de Lyon-Université Jean Monnet [Saint-Étienne] (UJM)-Institut National des Sciences Appliquées de Lyon (INSA Lyon), Université de Lyon-Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-Centre National de la Recherche Scientifique (CNRS), Department of Mathematics and Statistics [Ottawa], University of Ottawa [Ottawa], NSERC through a Discovery grant for the second author, ANR-10-LABX-0070,MILYON,Community of mathematics and fundamental computer science in Lyon(2010), ANR-15-CE40-0012,GéoLie,Méthodes géométriques en théorie de Lie(2015), Université de Lyon-Institut National des Sciences Appliquées de Lyon (INSA Lyon), Université de Lyon-Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-Université Jean Monnet [Saint-Étienne] (UJM)-Centre National de la Recherche Scientifique (CNRS), and University of Ottawa [Ottawa] (uOttawa)
Subjects: Springer Odd Degree Extension Theorem, 11E81, Rings and Algebras (math.RA), Springer Odd Degree Extension Theorem MSC 2000: 11E08, [MATH.MATH-RA]Mathematics [math]/Rings and Algebras [math.RA], FOS: Mathematics, Mathematics - Rings and Algebras, Quadratic forms, semilocal rings
Abstract: International audience; A fundamental result of Springer says that a quadratic form over a field of characteristic not 2 is isotropic if it is so after an odd degree extension. In this paper we generalize Springer's theorem as follows. Let R be a an arbitrary semilocal ring, let S be a finite R-algebra of constant odd degree, which is étale or generated by one element, and let q be a nonsingular R-quadratic form whose base ring extension q S is isotropic. We show that then q is already isotropic.
Published: 2021

26. Model theory of divisible abelian cyclically ordered groups and minimal C. O. G

Author: Leloup, Gérard, Laboratoire Manceau de Mathématiques (LMM), and Le Mans Université (UM)
Subjects: [MATH.MATH-LO]Mathematics [math]/Logic [math.LO], Mathematics::K-Theory and Homology, Rings and Algebras (math.RA), [MATH.MATH-RA]Mathematics [math]/Rings and Algebras [math.RA], FOS: Mathematics, Mathematics - Logic, Mathematics - Rings and Algebras, Logic (math.LO)
Abstract: We make available some results about model theory cyclically ordered groups. We start with a classification of complete theories of divisible abelian cyclically ordered groups. Most of this paper is dedicated to the study of the model theory of torsion-free divisible abelian cyclically ordered groups, since we first show that the theory of divisible abelian cyclically ordered groups with non-trivial torsion elements is complete. Then we look at the cyclically ordered groups where the only parametrically definable subsets are finite unions of singletons and open intervals, and those where the definable subsets are finite union of singletons and c-convex subsets, where being c-convex is the analogue of being convex in the linearly ordered case.
Published: 2021

27. A general construction of family algebraic structures

Author: Foissy, Loïc, Manchon, Dominique, Zhang, Yuanyuan, Laboratoire de Mathématiques Pures et Appliquées Joseph Liouville (LMPA), Université du Littoral Côte d'Opale (ULCO), Laboratoire de Mathématiques Blaise Pascal (LMBP), Université Blaise Pascal - Clermont-Ferrand 2 (UBP)-Centre National de la Recherche Scientifique (CNRS), Lanzhou University, and Foissy, Loïc
Subjects: graded objects, operads, colored operads, 2010 Mathematics Subject Classification: 16W99, 16S10, 08B20, 16T30, 17D25, typed decorated planar binary trees, [MATH.MATH-RA]Mathematics [math]/Rings and Algebras [math.RA], [MATH.MATH-RA] Mathematics [math]/Rings and Algebras [math.RA], duplicial algebras, diassociative semigroups, Monoidal categories, dendriform algebras, pre-Lie algebras
Abstract: 35 pages; We give a general account of family algebras over a finitely presented linear operad. In a family algebra, each operation of arity $n$ is replaced by a family of operations indexed by $\Omega^n$, where $\Omega$ is a set of parameters. We show that the operad, together with its presentation, naturally defines an algebraic structure on the set of parameters, which in turn is used in the description of the family version of the relations between operations. The examples of dendriform and duplicial family algebras (hence with two parameters) and operads are treated in detail, as well as the pre-Lie family case. Finally, free one-parameter duplicial family algebras are described, together with the extended duplicial semigroup structure on the set of parameters.
Published: 2021

28. On extended associative semigroups

Author: FOISSY, Loïc, Laboratoire de Mathématiques Pures et Appliquées Joseph Liouville (LMPA), and Université du Littoral Côte d'Opale (ULCO)
Subjects: braid equation, [MATH.MATH-RA]Mathematics [math]/Rings and Algebras [math.RA], diassociative semigroups, Mathematics - Rings and Algebras, Semigroups
Abstract: We study extended associative semigroups (briefly, EAS), an algebraic structure used to define generalizations of the operad of associative algebras, and the subclass of commutative extended diassociative semigroups (briefly, CEDS), which are used to define generalizations of the operad of pre-Lie algebras. We give families of examples based on semigroups or on groups, as well as a classification of EAS of cardinality two. We then define linear extended associative semigroups as linear maps satisfying a variation of the braid equation. We explore links between linear EAS and bialgebras and Hopf algebras. We also study the structure of nondegenerate finite CEDS and show that they are obtained by semidirect and direct products involving two groups.
Published: 2021

29. 2-adic parity explorations of the 3n+1 problem

Author: Laclé, Francis, Department of Information and Computing Sciences [Utrecht], Utrecht University [Utrecht], and Laclé, Francis
Subjects: [MATH.MATH-AC]Mathematics [math]/Commutative Algebra [math.AC], [INFO.INFO-DS]Computer Science [cs]/Data Structures and Algorithms [cs.DS], [MATH.MATH-DS]Mathematics [math]/Dynamical Systems [math.DS], Collatz Conjecture, [MATH.MATH-DS] Mathematics [math]/Dynamical Systems [math.DS], [INFO.INFO-DS] Computer Science [cs]/Data Structures and Algorithms [cs.DS], [INFO.INFO-DM]Computer Science [cs]/Discrete Mathematics [cs.DM], [MATH.MATH-GR]Mathematics [math]/Group Theory [math.GR], [MATH.MATH-AC] Mathematics [math]/Commutative Algebra [math.AC], [MATH.MATH-GM]Mathematics [math]/General Mathematics [math.GM], Number theory, [MATH.MATH-CO]Mathematics [math]/Combinatorics [math.CO], [MATH.MATH-RA] Mathematics [math]/Rings and Algebras [math.RA], Adic spaces, [MATH.MATH-GR] Mathematics [math]/Group Theory [math.GR], [INFO.INFO-MS]Computer Science [cs]/Mathematical Software [cs.MS], [INFO.INFO-AO]Computer Science [cs]/Computer Arithmetic, [MATH.MATH-RA]Mathematics [math]/Rings and Algebras [math.RA], [MATH.MATH-GM] Mathematics [math]/General Mathematics [math.GM], [INFO.INFO-NA]Computer Science [cs]/Numerical Analysis [cs.NA], [MATH.MATH-NT]Mathematics [math]/Number Theory [math.NT], [MATH.MATH-CO] Mathematics [math]/Combinatorics [math.CO], [INFO.INFO-DM] Computer Science [cs]/Discrete Mathematics [cs.DM], [INFO.INFO-MS] Computer Science [cs]/Mathematical Software [cs.MS], Arc diagrams, [INFO.INFO-NA] Computer Science [cs]/Numerical Analysis [cs.NA], [INFO.INFO-AO] Computer Science [cs]/Computer Arithmetic, Syracuse conjecture, [MATH.MATH-NT] Mathematics [math]/Number Theory [math.NT]
Abstract: Starting with a separate 2-adic reformulation of the Collatz function that maps parity orbits, we show that the stopping time $k$ of this reformulation can be alternatively derived with the help of some novel ideas as well as hybrid-algorithmic techniques.
Published: 2021

30. The Kostant invariant and special ε-orthogonal representations for ε-quadratic colour Lie algebras

Author: Philippe Meyer, Mathematical Institute [Oxford] (MI), University of Oxford [Oxford], Institut de Recherche Mathématique Avancée (IRMA), and Centre National de la Recherche Scientifique (CNRS)-Université de Strasbourg (UNISTRA)
Subjects: Pure mathematics, Algebra and Number Theory, [MATH.MATH-RT]Mathematics [math]/Representation Theory [math.RT], 010102 general mathematics, [MATH.MATH-RA]Mathematics [math]/Rings and Algebras [math.RA], Binary number, 01 natural sciences, Quadratic equation, 0103 physical sciences, Lie algebra, 010307 mathematical physics, 0101 mathematics, Invariant (mathematics), Mathematics
Abstract: International audience; Let $k$ be a field of characteristic not two or three, let $\mathfrak{g}$ be a finite-dimensional colour Lie algebra and let V be a finite-dimensional representation of $\mathfrak{g}$. In this article we give various ways of constructing a colour Lie algebra $\tilde{\mathfrak{g}}$ whose bracket in some sense extends both the bracket of $\mathfrak{g}$ and the action of $\mathfrak{g}$ on V. Colour Lie algebras, originally introduced by R. Ree ([Ree60]), generalise both Lie algebras and Lie superalgebras, and in those cases our results imply many known results ([Kos99], [Kos01], [CK15], [SS15]). For a class of representations arising in this context we show there are covariants satisfying identities analogous to Mathews identities for binary cubics.
Published: 2021
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31. Generalized associative algebras

Author: Foissy, Loïc, Laboratoire de Mathématiques Pures et Appliquées Joseph Liouville (LMPA), and Université du Littoral Côte d'Opale (ULCO)
Subjects: matching associative algebras, Mathematics::K-Theory and Homology, Mathematics::Quantum Algebra, Mathematics::Category Theory, [MATH.MATH-RA]Mathematics [math]/Rings and Algebras [math.RA], Mathematics - Rings and Algebras, Family associative algebras, two-parameters associative algebras, associative semigroups, Mathematics::Algebraic Topology
Abstract: We study diverse parametrized versions of the operad of associative algebra, where the parameter are taken in an associative semigroup $\Omega$ (generalization of matching or family associative algebras) or in its cartesian square (two-parameters associative algebras). We give a description of the free algebras on these operads, study their formal series and prove that they are Koszul when the set of parameters is finite. We also study operadic morphisms between the operad of classical associative algebras and these objects, and links with other types of algebras (diassociative, dendriform, post-Lie...).
Published: 2021

32. Algorithms for fundamental invariants and equivariants: (of finite groups)

Author: HUBERT, Evelyne, RODRIGUEZ BAZAN, Erick, AlgebRe, geOmetrie, Modelisation et AlgoriTHmes (AROMATH), Inria Sophia Antipolis - Méditerranée (CRISAM), Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-National and Kapodistrian University of Athens (NKUA), and Université Côte d'Azur (UCA)
Subjects: [INFO.INFO-SC]Computer Science [cs]/Symbolic Computation [cs.SC], symmetry adapted bases, equivariants, Covariants, [MATH.MATH-RA]Mathematics [math]/Rings and Algebras [math.RA], representation theory, polynomial invariants, H-basis. MSC: 13A50 20C30 68W30 13P10, H-basis, computational invariant theory, finite groups
Abstract: International audience; For a finite group, we present three algorithms to compute a generating set of invariant simultaneously to generating sets of basic equivariants, i.e., equivariants for the irreducible representations of the group. The main novelty resides in the exploitation of the orthogonal complement of the ideal generated by invariants; Its symmetry adapted basis delivers the fundamental equivariants. Fundamental equivariants allow to assemble symmetry adapted bases of polynomial spaces of higher degrees, and these are essential ingredients in exploiting and preserving symmetry in computations. They appear within algebraic computation and beyond, in physics, chemistry and engineering. Our first construction applies solely to reflection groups and consists in applying symmetry preserving interpolation, as developed by the same authors, along an orbit in general position. The fundamental invariants can be read off the H-basis of the ideal of the orbit while the fundamental equivariants are obtained from a symmetry adapted basis of an invariant direct complement to this ideal in the polynomial ring. The second algorithm takes as input primary invariants and the output provides not only the secondary invariants but also free bases for the modules of basic equivariants. These are constructed as the components of a symmetry adapted basis of the orthogonal complement, in the polynomial ring, to the ideal generated by primary invariants.The third algorithm proceeds degree by degree, determining the fundamental invariants as forming a H-basis of the Hilbert ideal, i.e., the polynomial ideal generated by the invariants of positive degree. The fundamental equivariants are simultaneously computed degree by degree as the components of a symmetry adapted basis of the orthogonal complement of the Hilbert ideal.
Published: 2021
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33. Generalized prelie and permutative algebras

Author: Foissy, Loïc, Laboratoire de Mathématiques Pures et Appliquées Joseph Liouville (LMPA), and Université du Littoral Côte d'Opale (ULCO)
Subjects: permutative algebras, Mathematics::Rings and Algebras, [MATH.MATH-RA]Mathematics [math]/Rings and Algebras [math.RA], typed rooted trees, Pre-Lie algebras, interacting bialgebras, Mathematics - Rings and Algebras
Abstract: We study generalizations of pre-Lie algebras, where the free objects are based on rooted trees which edges are typed, instead of usual rooted trees, and with generalized pre-Lie products formed by graftings. Working with a discrete set of types, we show how to obtain such objects when this set is given an associative commutative product and a second product making it a commutative extended semigroup. Working with a vector space of types, these two products are replaced by a bilinear map $\Phi$ which satisfies a braid equation and a commutation relation. Examples of such structures are defined on sets, semigroups, or groups. These constructions define a family of operads PreLie $\Phi$ which generalize the operad of pre-Lie algebras PreLie. For any embedding from PreLie into PreLie $\phi$ , we construct a family of pairs of cointeracting bialgebras, based on typed and decorated trees: the first coproduct is given by an extraction and contraction process, the types being modified by the action of $\Phi$; the second coproduct is given by admissible cuts, in the Connes and Kreimer's way, with again types modified by the action of $\Phi$. We also study the Koszul dual of PreLie $\Phi$ , which gives generalizations of permutative algebras.
Published: 2021

34. n-CHAINED SEMIGROUPS AND n/2-PERSPECTIVE MODULES AND RINGS

Author: Mary, Xavier, Modélisation aléatoire de Paris X (MODAL'X), Université Paris Nanterre (UPN), Fédération Parisienne de Modélisation Mathématique (FP2M), Centre National de la Recherche Scientifique (CNRS), ANR-11-LABX-0023,MME-DII,Modèles Mathématiques et Economiques de la Dynamique, de l'Incertitude et des Interactions(2011), Mary, Xavier, and Modèles Mathématiques et Economiques de la Dynamique, de l'Incertitude et des Interactions - - MME-DII2011 - ANR-11-LABX-0023 - LABX - VALID
Subjects: (special) clean elements, associate idempotents, [MATH.MATH-RA]Mathematics [math]/Rings and Algebras [math.RA], [MATH.MATH-RA] Mathematics [math]/Rings and Algebras [math.RA], reflexive inverses, perspective modules and rings, (strongly) regular elements and rings, 15A09, 16D70, 16E50, 16U99, 20M99, semigroups, [MATH.MATH-GR]Mathematics [math]/Group Theory [math.GR], [MATH.MATH-GR] Mathematics [math]/Group Theory [math.GR]
Abstract: We introduce and study a new class of modules and rings we call n/2-perspective, which can either be described in terms of perspective direct summands, associate idempotents, or generalized inverses. When n is small (≤ 2), we recover existing class of modules and rings: (endo)abelian, strongly IC and perspective ones. And 3/2-perspective rings are characterized by all their regular elements being special clean. Standard constructions are also discussed and examples are provided.
Published: 2021
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35. (b, c)-INVERSE, INVERSE ALONG AN ELEMENT, AND THE SCHUTZENBERGER CATEGORY OF A SEMIGROUP

Author: Xavier Mary, Modélisation aléatoire de Paris X (MODAL'X), Université Paris Nanterre (UPN), Fédération Parisienne de Modélisation Mathématique (FP2M), Centre National de la Recherche Scientifique (CNRS), ANR-11-LABX-0023,MME-DII,Modèles Mathématiques et Economiques de la Dynamique, de l'Incertitude et des Interactions(2011), Mary, Xavier, and Modèles Mathématiques et Economiques de la Dynamique, de l'Incertitude et des Interactions - - MME-DII2011 - ANR-11-LABX-0023 - LABX - VALID
Subjects: Pure mathematics, generalized inverses, Semigroup, Mathematics::Operator Algebras, Applied Mathematics, [MATH.MATH-RA]Mathematics [math]/Rings and Algebras [math.RA], Inverse, 16. Peace & justice, reverse order law, [MATH.MATH-GR]Mathematics [math]/Group Theory [math.GR], Reverse order, Computational Mathematics, Morphism, Mathematics::Category Theory, [MATH.MATH-RA] Mathematics [math]/Rings and Algebras [math.RA], QA1-939, Discrete Mathematics and Combinatorics, Element (category theory), Analysis, green’s relations, Mathematics, [MATH.MATH-GR] Mathematics [math]/Group Theory [math.GR]
Abstract: In this short note, we study two specific generalized inverses, the (b, c)-inverse of Drazin and the inverse along an element of the present author, in a categorical way. Precisely, we prove that the (b, c)-inverse and the inverse along an element in a semigroup are actually genuine inverse when considered as morphisms in the Schutzenberger category of the semigroup. This category was first defined and studied by Costa and Steinberg, and its definition is based on Green's preoders and relations. In the last section, applications to the Reverse Order Law are given and some new formulas are provided. Examples are also given.
Published: 2021
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36. When is a reductive group scheme linear?

Author: Gille, Philippe, Institut Camille Jordan [Villeurbanne] (ICJ), École Centrale de Lyon (ECL), Université de Lyon-Université de Lyon-Université Claude Bernard Lyon 1 (UCBL), Université de Lyon-Institut National des Sciences Appliquées de Lyon (INSA Lyon), Université de Lyon-Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-Université Jean Monnet [Saint-Étienne] (UJM)-Centre National de la Recherche Scientifique (CNRS), Université de Lyon-Université Jean Monnet [Saint-Étienne] (UJM)-Institut National des Sciences Appliquées de Lyon (INSA Lyon), Université de Lyon-Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-Centre National de la Recherche Scientifique (CNRS), Institut National des Sciences Appliquées (INSA)-Université de Lyon-Institut National des Sciences Appliquées (INSA)-Université Jean Monnet [Saint-Étienne] (UJM)-Centre National de la Recherche Scientifique (CNRS), Centre National de la Recherche Scientifique (CNRS)-Université Jean Monnet [Saint-Étienne] (UJM)-École Centrale de Lyon (ECL), Université de Lyon-Université de Lyon-Institut National des Sciences Appliquées de Lyon (INSA Lyon), Université de Lyon-Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-Université Claude Bernard Lyon 1 (UCBL), and Université de Lyon
Subjects: General Mathematics, [MATH.MATH-RA]Mathematics [math]/Rings and Algebras [math.RA], 20G35, representations, Mathematics - Rings and Algebras, Mathematics - Algebraic Geometry, resolution property MSC 2000: 14L15, tori, Rings and Algebras (math.RA), FOS: Mathematics, resolution property, [MATH.MATH-AG]Mathematics [math]/Algebraic Geometry [math.AG], Reductive group schemes, Algebraic Geometry (math.AG)
Abstract: International audience; We show that a reductive group scheme over a base scheme S admits a faithful linear representation if and only if its radical torus is isotrivial, that is, it splits after a finite étale cover.
Published: 2021

37. Constructive basic theory of central simple algebras

Author: Coquand, Thierry, Lombardi, Henri, Neuwirth, Stefan, Laboratoire de Mathématiques de Besançon (UMR 6623) (LMB), Université de Bourgogne (UB)-Université de Franche-Comté (UFC), Université Bourgogne Franche-Comté [COMUE] (UBFC)-Université Bourgogne Franche-Comté [COMUE] (UBFC)-Centre National de la Recherche Scientifique (CNRS), and ANR-19-CE40-0002,ANCG,Analyse non commutative sur les groupes et les groupes quantiques(2019)
Subjects: 16B70, 03F65, 19C30, MSC 16B70 (03F65 19C30), Rings and Algebras (math.RA), [MATH.MATH-RA]Mathematics [math]/Rings and Algebras [math.RA], FOS: Mathematics, Mathematics - Rings and Algebras, Mathematics - Logic, Logic (math.LO)
Abstract: We provide a constructive treatment of basic results in the theory of central simple algebras. One main issue is the fact that one starting result, Wedderburn's Theorem stating that a simple algebra is a matrix algebra over a skew field, is not constructively valid. We solve this problem by proving instead a dynamical version of this theorem. One can use this to give constructive proofs of basic results of the theory of central simple algebras, such as the Skolem-Noether Theorem. We illustrate this development by giving an elementary constructive proof of a theorem of Becher (which is itself a consequence of a celebrated theorem of Merkurjev).
Published: 2021

38. Unsolvability of the Quintic Formalized in Dependent Type Theory

Author: Bernard, Sophie, Cohen, Cyril, Mahboubi, Assia, Strub, Pierre Yves, Cohen, Liron, Kaliszyk, Cezary, Sûreté du logiciel et Preuves Mathématiques Formalisées (STAMP), Inria Sophia Antipolis - Méditerranée (CRISAM), Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria), Université Côte d'Azur (UCA), Laboratoire des Sciences du Numérique de Nantes (LS2N), Université de Nantes - Faculté des Sciences et des Techniques, Université de Nantes (UN)-Université de Nantes (UN)-École Centrale de Nantes (ECN)-Centre National de la Recherche Scientifique (CNRS)-IMT Atlantique Bretagne-Pays de la Loire (IMT Atlantique), Institut Mines-Télécom [Paris] (IMT)-Institut Mines-Télécom [Paris] (IMT), Gallinette : vers une nouvelle génération d'assistant à la preuve (GALLINETTE), Inria Rennes – Bretagne Atlantique, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-Laboratoire des Sciences du Numérique de Nantes (LS2N), Institut Mines-Télécom [Paris] (IMT)-Institut Mines-Télécom [Paris] (IMT)-Université de Nantes - Faculté des Sciences et des Techniques, Laboratoire d'informatique de l'École polytechnique [Palaiseau] (LIX), Centre National de la Recherche Scientifique (CNRS)-École polytechnique (X), IMT Atlantique Bretagne-Pays de la Loire (IMT Atlantique), Institut Mines-Télécom [Paris] (IMT)-Institut Mines-Télécom [Paris] (IMT)-Université de Nantes - UFR des Sciences et des Techniques (UN UFR ST), Université de Nantes (UN)-Université de Nantes (UN)-École Centrale de Nantes (ECN)-Centre National de la Recherche Scientifique (CNRS), Vrije Universiteit Amsterdam [Amsterdam] (VU), École polytechnique (X)-Centre National de la Recherche Scientifique (CNRS), Cohen, Liron, Kaliszyk, Cezary, Mathematics, Gallinette : vers une nouvelle génération d'assistant à la preuve (LS2N - équipe GALLINETTE), Université de Nantes - UFR des Sciences et des Techniques (UN UFR ST), and Université de Nantes (UN)-Université de Nantes (UN)-École Centrale de Nantes (ECN)-Centre National de la Recherche Scientifique (CNRS)-IMT Atlantique (IMT Atlantique)
Subjects: General quintic, Galois theory, [MATH.MATH-RA]Mathematics [math]/Rings and Algebras [math.RA], Mathematical Components, [INFO.INFO-LO]Computer Science [cs]/Logic in Computer Science [cs.LO], Theory of computation → Type theory, Theory of computation → Logic and verification, [MATH.MATH-GR]Mathematics [math]/Group Theory [math.GR], TheoryofComputation_MATHEMATICALLOGICANDFORMALLANGUAGES, Dependent Type Theory, ComputingMethodologies_SYMBOLICANDALGEBRAICMANIPULATION, Coq, Theory of computation → Constructive mathematics, Abel-Ruffini
Abstract: In this paper, we describe an axiom-free Coq formalization that there does not exists a general method for solving by radicals polynomial equations of degree greater than 4. This development includes a proof of Galois' Theorem of the equivalence between solvable extensions and extensions solvable by radicals. The unsolvability of the general quintic follows from applying this theorem to a well chosen polynomial with unsolvable Galois group., LIPIcs, Vol. 193, 12th International Conference on Interactive Theorem Proving (ITP 2021), pages 8:1-8:18
Published: 2021
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39. On the monoidal invariance of the cohomological dimension of Hopf algebras

Author: Julien Bichon, Laboratoire de Mathématiques Blaise Pascal (LMBP), and Centre National de la Recherche Scientifique (CNRS)-Université Clermont Auvergne (UCA)
Subjects: General Mathematics, Mathematics::Rings and Algebras, [MATH.MATH-RA]Mathematics [math]/Rings and Algebras [math.RA], K-Theory and Homology (math.KT), Mathematics - Rings and Algebras, Rings and Algebras (math.RA), Mathematics::K-Theory and Homology, Mathematics::Category Theory, Mathematics::Quantum Algebra, Mathematics - K-Theory and Homology, Mathematics - Quantum Algebra, FOS: Mathematics, [MATH.MATH-QA]Mathematics [math]/Quantum Algebra [math.QA], [MATH.MATH-KT]Mathematics [math]/K-Theory and Homology [math.KT], Quantum Algebra (math.QA)
Abstract: We discuss the question of whether the global dimension is a monoidal invariant for Hopf algebras, in the sense that if two Hopf algebras have equivalent monoidal categories of comodules, then their global dimensions should be equal. We provide several positive new answers to this question, under various assumptions of smoothness, cosemisimplicity or finite dimension. We also discuss the comparison between the global dimension and the Gerstenhaber-Schack cohomological dimension in the cosemisimple case, obtaining equality in the case the latter is finite. One of our main tools is the new concept of twisted separable functor., 20 pages. V2: 23 pages, changes in the redaction, no new result
Published: 2021

40. Sur les Initiaux et le Théorème Fondamental de la Géométrie Différentielle Tropicale Partielle

Author: Falkensteiner, Sebastian, Garay-Lopez, Cristhian Emmanuel, Haiech, Mercedes, Noordman, Marc Paul, Boulier, François, Toghani, Zeinab, Research Institute for Symbolic Computation (RISC), Johannes Kepler Universität Linz (JKU), Centro de Investigación en Matemáticas (CIMAT), Consejo Nacional de Ciencia y Tecnología [Mexico] (CONACYT), Institut de Recherche Mathématique de Rennes (IRMAR), Université de Rennes 1 (UR1), Université de Rennes (UNIV-RENNES)-Université de Rennes (UNIV-RENNES)-AGROCAMPUS OUEST, Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro)-Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro)-Institut National des Sciences Appliquées - Rennes (INSA Rennes), Institut National des Sciences Appliquées (INSA)-Université de Rennes (UNIV-RENNES)-Institut National des Sciences Appliquées (INSA)-École normale supérieure - Rennes (ENS Rennes)-Université de Rennes 2 (UR2), Université de Rennes (UNIV-RENNES)-Centre National de la Recherche Scientifique (CNRS), Bernoulli Institute for Mathematics and Computer Science and Artificial Intelligence, University of Groningen [Groningen], Centre de Recherche en Informatique, Signal et Automatique de Lille - UMR 9189 (CRIStAL), Université de Lille-Ecole Centrale de Lille-Centre National de la Recherche Scientifique (CNRS), School of Mathematical Sciences [Queen Mary], and University of London [London]
Subjects: [INFO.INFO-SC]Computer Science [cs]/Symbolic Computation [cs.SC], [MATH.MATH-AC]Mathematics [math]/Commutative Algebra [math.AC], [MATH.MATH-RA]Mathematics [math]/Rings and Algebras [math.RA]
Published: 2021

41. Intrinsic approach to Galois theory of q-difference equations: with a preface to Part IV by Anne Granier

Author: Hardouin, Charlotte, Di Vizio, Lucia, Institut de Mathématiques de Toulouse UMR5219 (IMT), Institut National des Sciences Appliquées - Toulouse (INSA Toulouse), Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-Université Toulouse 1 Capitole (UT1), Université Fédérale Toulouse Midi-Pyrénées-Université Fédérale Toulouse Midi-Pyrénées-Université Toulouse - Jean Jaurès (UT2J)-Université Toulouse III - Paul Sabatier (UT3), Université Fédérale Toulouse Midi-Pyrénées-Centre National de la Recherche Scientifique (CNRS), Laboratoire de Mathématiques de Versailles (LMV), Université de Versailles Saint-Quentin-en-Yvelines (UVSQ)-Université Paris-Saclay-Centre National de la Recherche Scientifique (CNRS), Université Toulouse Capitole (UT Capitole), Université de Toulouse (UT)-Université de Toulouse (UT)-Institut National des Sciences Appliquées - Toulouse (INSA Toulouse), Institut National des Sciences Appliquées (INSA)-Université de Toulouse (UT)-Institut National des Sciences Appliquées (INSA)-Université Toulouse - Jean Jaurès (UT2J), Université de Toulouse (UT)-Université Toulouse III - Paul Sabatier (UT3), Université de Toulouse (UT)-Centre National de la Recherche Scientifique (CNRS), ANR-06-JCJC-0028,DIOPHANTE,Approche géométrique à l'approximation diophantienne(2006), and ANR-10-JCJC-0105,Q-DIFF,Aspects algébriques et analytiques des équations aux q-différences(2010)
Subjects: [MATH.MATH-RA]Mathematics [math]/Rings and Algebras [math.RA], [MATH]Mathematics [math], ComputingMilieux_MISCELLANEOUS, [MATH.MATH-NT]Mathematics [math]/Number Theory [math.NT]
Abstract: International audience
Published: 2021
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42. Formal deformations of the algebra of Jacobi forms and Rankin-Cohen brackets

Author: Choie, Youngju, Dumas, François, Martin, François, Royer, Emmanuel, Department of Mathematics, POSTECH, Pohang University of Science and Technology (POSTECH), Laboratoire de Mathématiques Blaise Pascal (LMBP), Centre National de la Recherche Scientifique (CNRS)-Université Clermont Auvergne (UCA), and NRF 2017R1A2B2001807 of the National Research Fund of KoreaCAP2025 de l'Université Clermont Auvergne
Subjects: Poisson brackets, MSC 53D55, 17B63, 11F25, 11F11, 16W25, quasimodular forms, Serre derivative, [MATH.MATH-RA]Mathematics [math]/Rings and Algebras [math.RA], Jacobi forms, formal deformation, star product, Rankin-Cohen brackets, Connes-Moscovici Theorem, [MATH.MATH-NT]Mathematics [math]/Number Theory [math.NT]
Abstract: International audience; This work is devoted to the algebraic and arithmetic properties of Rankin-Cohen brackets allowing to define and study them in several natural situations of number theory. It focuses on the property of these brackets to be formal deformations of the algebras on which they are defined, with related questions on restriction-extension methods. The general algebraic results developed here are applied to the study of formal deformations of the algebra of weak Jacobi forms and their relation with the Rankin-Cohen brackets on modular and quasimodular forms.; Ce travail est consacré aux propriétés algébriques et arithmétiques des crochets de Rankin-Cohen permettant de les définir et de les étudier dans plusieurs situations naturelles de la théorie des nombres. Il se concentre sur la propriété qu'ont ces crochets d'être des déformations formelles des algèbres sur lesquelles ils sont définis, avec des questions connexes sur les méthodes de restriction-extension. Les résultats algébriques généraux développés ici sont appliqués à l'étude des déformations formelles de l'algèbre des formes de Jacobi faibles et leur relation avec les crochets de Rankin-Cohen pour les formes modulaires et quasimodulaires.
Published: 2021
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43. Typed binary trees and generalized dendrifom algebras

Author: Loïc Foissy, Foissy, Loïc, Laboratoire de Mathématiques Pures et Appliquées Joseph Liouville (LMPA), and Université du Littoral Côte d'Opale (ULCO)
Subjects: Matching (graph theory), Structure (category theory), 02 engineering and technology, 01 natural sciences, Set (abstract data type), Combinatorics, 05C05, Cardinality, 18D50, Mathematics::Category Theory, Mathematics::Quantum Algebra, Dendriform algebra, 0202 electrical engineering, electronic engineering, information engineering, [MATH.MATH-RA] Mathematics [math]/Rings and Algebras [math.RA], Order (group theory), shuffle product AMS classification 16T30, 0101 mathematics, Axiom, Mathematics, diassociative semigroup, Algebra and Number Theory, Binary tree, Semigroup, 010102 general mathematics, [MATH.MATH-RA]Mathematics [math]/Rings and Algebras [math.RA], Mathematics::Rings and Algebras, plane binary trees, 020201 artificial intelligence & image processing
Abstract: We here both unify and generalize nonassociative structures on typed binary trees, that is to say plane binary trees which edges are decorated by elements of a set Ω. We prove that we obtain such a structure, called an Ω-dendriform structure, if Ω has four products satisfying certain axioms (EDS axioms), including the axioms of a diassociative semigroup. This includes matching dendriform algebras introduced by Zhang, Gao and Guo and family dendriform algebras associated to a semigroup introduced by Zhang, Gao and Manchon, and of course dendriform algebras when Ω is reduced to a single element. We also give examples of EDS, including all the EDS of cardinality two; a combinatorial description of the products of such a structure on typed binary trees, but also on words; a study of the Koszul dual of the associated operads; and considerations on the existence of a coproduct, in order to obtain dendriform bialgebras.
Published: 2021

44. Operads of enriched pre-Lie algebras and freeness theorems

Author: Dotsenko, Vladimir, Foissy, Loïc, Institut de Recherche Mathématique Avancée (IRMA), Centre National de la Recherche Scientifique (CNRS)-Université de Strasbourg (UNISTRA), Laboratoire de Mathématiques Pures et Appliquées Joseph Liouville (LMPA), and Université du Littoral Côte d'Opale (ULCO)
Subjects: Mathematics::K-Theory and Homology, Mathematics::Quantum Algebra, Mathematics::Category Theory, [MATH.MATH-RA]Mathematics [math]/Rings and Algebras [math.RA], [MATH]Mathematics [math], Mathematics::Algebraic Topology, [MATH.MATH-CT]Mathematics [math]/Category Theory [math.CT]
Abstract: International audience; In this paper, we study the operad of C-enriched pre-Lie algebras, defined for any Hopf cooperad C; it slightly generalises a similar notion defined by Calaque and Willwacher to produce conceptual constructions of the operads acting on various deformation complexes. Maps between Hopf cooperads lead to maps between the corresponding operads of enriched pre-Lie algebras; we prove a criterion for the module action of the domain on the codomain to be free, on the left and on the right. In particular, this implies a new functorial Poincaré--Birkhoff--Witt type theorem for universal enveloping brace algebras of pre-Lie algebras.
Published: 2021

45. The universal pre-Lie-Rinehart algebras of aromatic trees

Author: Fløystad, Gunnar, Manchon, Dominique, Munthe-Kaas, Hans Z., Department of Mathematics [Bergen] (UiB), University of Bergen (UiB), Laboratoire de Mathématiques Blaise Pascal (LMBP), Université Clermont Auvergne [2017-2020] (UCA [2017-2020])-Centre National de la Recherche Scientifique (CNRS), and Ali Baklouti, Hideyuki Ishi
Subjects: Mathematics::K-Theory and Homology, [MATH.MATH-RA]Mathematics [math]/Rings and Algebras [math.RA], [MATH]Mathematics [math]
Abstract: 15 pages, 1 figure; International audience; We organize colored aromatic trees into a pre-Lie-Rinehart algebra (i.e. a flat torsion-free Lie-Rinehart algebra) endowed with a natural trace map, and show the freeness of this object among pre-Lie-Rinehart algebras with trace. This yields the algebraic foundations of aromatic B-series.
Published: 2021
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46. Rank-metric codes over arbitrary Galois extensions and rank analogues of Reed-Muller codes

Author: Alain Couvreur, Alessandro Neri, Daniel Augot, Julien Lavauzelle, Geometry, arithmetic, algorithms, codes and encryption (GRACE), Laboratoire d'informatique de l'École polytechnique [Palaiseau] (LIX), Centre National de la Recherche Scientifique (CNRS)-École polytechnique (X)-Centre National de la Recherche Scientifique (CNRS)-École polytechnique (X)-Inria Saclay - Ile de France, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria), Institut de Recherche Mathématique de Rennes (IRMAR), AGROCAMPUS OUEST, Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro)-Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro)-Université de Rennes 1 (UR1), Université de Rennes (UNIV-RENNES)-Université de Rennes (UNIV-RENNES)-Université de Rennes 2 (UR2), Université de Rennes (UNIV-RENNES)-École normale supérieure - Rennes (ENS Rennes)-Centre National de la Recherche Scientifique (CNRS)-Institut National des Sciences Appliquées - Rennes (INSA Rennes), Institut National des Sciences Appliquées (INSA)-Université de Rennes (UNIV-RENNES)-Institut National des Sciences Appliquées (INSA), Technische Universität Munchen - Université Technique de Munich [Munich, Allemagne] (TUM), ANR-15-CE39-0013,Manta,Geométrie algébrique et théorie des codes pour la cryptographie(2015), École polytechnique (X)-Centre National de la Recherche Scientifique (CNRS)-École polytechnique (X)-Centre National de la Recherche Scientifique (CNRS)-Inria Saclay - Ile de France, Université de Rennes (UR)-Institut National des Sciences Appliquées - Rennes (INSA Rennes), Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-École normale supérieure - Rennes (ENS Rennes)-Université de Rennes 2 (UR2)-Centre National de la Recherche Scientifique (CNRS)-INSTITUT AGRO Agrocampus Ouest, Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro)-Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro), French Direction Generale l'Armement through the Pole d'excellence cyber, 187711, Swiss National Science Foundation, ANR-15-CE39-0013, Agence Nationale de la Recherche, and ANR-11-LABX-0020,LEBESGUE,Centre de Mathématiques Henri Lebesgue : fondements, interactions, applications et Formation(2011)
Subjects: FOS: Computer and information sciences, Rank (linear algebra), Computer Science - Information Theory, Galois theory, Galois group, 010103 numerical & computational mathematics, 01 natural sciences, FOS: Mathematics, Number Theory (math.NT), 0101 mathematics, [MATH]Mathematics [math], Mathematics, Discrete mathematics, Algebra and Number Theory, Mathematics - Number Theory, Group (mathematics), Information Theory (cs.IT), Applied Mathematics, 010102 general mathematics, [MATH.MATH-RA]Mathematics [math]/Rings and Algebras [math.RA], Skew, Reed–Muller code, [MATH.MATH-IT]Mathematics [math]/Information Theory [math.IT], Mathematics - Rings and Algebras, Linear subspace, [MATH.MATH-NT]Mathematics [math]/Number Theory [math.NT], Rings and Algebras (math.RA), Metric (mathematics), Geometry and Topology
Abstract: This paper extends the study of rank-metric codes in extension fields $\mathbb{L}$ equipped with an arbitrary Galois group $G = \mathrm{Gal}(\mathbb{L}/\mathbb{K})$. We propose a framework for studying these codes as subspaces of the group algebra $\mathbb{L}[G]$, and we relate this point of view with usual notions of rank-metric codes in $\mathbb{L}^N$ or in $\mathbb{K}^{N\times N}$, where $N = [\mathbb{L} : \mathbb{K}]$. We then adapt the notion of error-correcting pairs to this context, in order to provide a non-trivial decoding algorithm for these codes. We then focus on the case where $G$ is abelian, which leads us to see codewords as elements of a multivariate skew polynomial ring. We prove that we can bound the dimension of the vector space of zeroes of these polynomials, depending of their degree. This result can be seen as an analogue of Alon-F\"uredi theorem -- and by means, of Schwartz-Zippel lemma -- in the rank metric. Finally, we construct the counterparts of Reed-Muller codes in the rank metric, and we give their parameters. We also show the connection between these codes and classical Reed-Muller codes in the case where $\mathbb{L}$ is a Kummer extension., Comment: 26 pages, 1 figure
Published: 2021
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47. Homogeneous localizations of some quantum enveloping superalgebras

Author: Jacques Alev, François Dumas, Laboratoire de Mathématiques de Reims (LMR), Université de Reims Champagne-Ardenne (URCA)-Centre National de la Recherche Scientifique (CNRS), Laboratoire de Mathématiques Blaise Pascal (LMBP), and Université Clermont Auvergne [2017-2020] (UCA [2017-2020])-Centre National de la Recherche Scientifique (CNRS)
Subjects: Noetherian, Pure mathematics, Algebra and Number Theory, Applied Mathematics, 010102 general mathematics, Mathematics::Rings and Algebras, [MATH.MATH-RA]Mathematics [math]/Rings and Algebras [math.RA], Lie superalgebra, 01 natural sciences, Superalgebra, Homogeneous, Mathematics::Quantum Algebra, 0103 physical sciences, [MATH.MATH-QA]Mathematics [math]/Quantum Algebra [math.QA], 010307 mathematical physics, 0101 mathematics, Equivalence (formal languages), Mathematics::Representation Theory, Quantum, Mathematics
Abstract: International audience; Under suitable conditions the skewfield of fractions of a su-peralgebra which is a noetherian domain is canonically provided with a structure of superalgebra. This gives rise to a notion of rational equivalence in the category of superalgebras. We study from the point of view of this rational equivalence some low dimensional examples of quantum enveloping algebras of Lie superalgebras.
Published: 2021
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48. A theory of residues for skew rational functions

Author: Xavier Caruso, Institut de Mathématiques de Bordeaux (IMB), Université Bordeaux Segalen - Bordeaux 2-Université Sciences et Technologies - Bordeaux 1-Université de Bordeaux (UB)-Institut Polytechnique de Bordeaux (Bordeaux INP)-Centre National de la Recherche Scientifique (CNRS), Lithe and fast algorithmic number theory (LFANT), Université Bordeaux Segalen - Bordeaux 2-Université Sciences et Technologies - Bordeaux 1-Université de Bordeaux (UB)-Institut Polytechnique de Bordeaux (Bordeaux INP)-Centre National de la Recherche Scientifique (CNRS)-Université Bordeaux Segalen - Bordeaux 2-Université Sciences et Technologies - Bordeaux 1-Université de Bordeaux (UB)-Institut Polytechnique de Bordeaux (Bordeaux INP)-Centre National de la Recherche Scientifique (CNRS)-Inria Bordeaux - Sud-Ouest, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria), Université Bordeaux Segalen - Bordeaux 2-Université Sciences et Technologies - Bordeaux 1 (UB)-Université de Bordeaux (UB)-Institut Polytechnique de Bordeaux (Bordeaux INP)-Centre National de la Recherche Scientifique (CNRS), and Université Bordeaux Segalen - Bordeaux 2-Université Sciences et Technologies - Bordeaux 1 (UB)-Université de Bordeaux (UB)-Institut Polytechnique de Bordeaux (Bordeaux INP)-Centre National de la Recherche Scientifique (CNRS)-Université Bordeaux Segalen - Bordeaux 2-Université Sciences et Technologies - Bordeaux 1 (UB)-Université de Bordeaux (UB)-Institut Polytechnique de Bordeaux (Bordeaux INP)-Centre National de la Recherche Scientifique (CNRS)-Inria Bordeaux - Sud-Ouest
Subjects: [INFO.INFO-SC]Computer Science [cs]/Symbolic Computation [cs.SC], Pure mathematics, Change of variables, Skew polynomials, General Mathematics, 010102 general mathematics, [MATH.MATH-RA]Mathematics [math]/Rings and Algebras [math.RA], Skew, [MATH.MATH-IT]Mathematics [math]/Information Theory [math.IT], Rational function, 01 natural sciences, 0103 physical sciences, 010307 mathematical physics, 0101 mathematics, Quotient, Mathematics
Abstract: International audience; This paper constitutes a first attempt to do analysis with skew polynomials. Precisely, our main objective is to develop a theory of residues for skew rational functions (which are, by definition, the quotients of two skew polynomials). We prove in particular a skew analogue of the residue formula and a skew analogue of the classical formula of change of variables for residues.
Published: 2021
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49. Shuffle polygraphic resolutions for operads

Author: Malbos, Philippe, Ren, Isaac, Algèbre, géométrie, logique (AGL), Institut Camille Jordan (ICJ), École Centrale de Lyon (ECL), Université de Lyon-Université de Lyon-Université Claude Bernard Lyon 1 (UCBL), Université de Lyon-Institut National des Sciences Appliquées de Lyon (INSA Lyon), Université de Lyon-Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-Université Jean Monnet - Saint-Étienne (UJM)-Centre National de la Recherche Scientifique (CNRS)-École Centrale de Lyon (ECL), Université de Lyon-Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-Université Jean Monnet - Saint-Étienne (UJM)-Centre National de la Recherche Scientifique (CNRS), École normale supérieure - Lyon (ENS Lyon), Institut Camille Jordan [Villeurbanne] (ICJ), Université de Lyon-Université Jean Monnet [Saint-Étienne] (UJM)-Institut National des Sciences Appliquées de Lyon (INSA Lyon), Université de Lyon-Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-Centre National de la Recherche Scientifique (CNRS)-École Centrale de Lyon (ECL), and Université de Lyon-Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-Centre National de la Recherche Scientifique (CNRS)
Subjects: Higher-dimensional rewriting, General Mathematics, [MATH.MATH-RA]Mathematics [math]/Rings and Algebras [math.RA], Mathematics - Category Theory, K-Theory and Homology (math.KT), Mathematics::Algebraic Topology, Mathematics::K-Theory and Homology, Mathematics::Quantum Algebra, Computer Science::Logic in Computer Science, Mathematics::Category Theory, Koszulness, Mathematics - K-Theory and Homology, FOS: Mathematics, [MATH.MATH-KT]Mathematics [math]/K-Theory and Homology [math.KT], Category Theory (math.CT), Gröbner bases, Shuffle operads, 18M70, 68Q42, 18N30, [MATH.MATH-CT]Mathematics [math]/Category Theory [math.CT]
Abstract: Shuffle operads were introduced to forget the symmetric group actions on symmetric operads while preserving all possible operadic compositions. Rewriting methods were then applied to symmetric operads via shuffle operads: in particular, a notion of Gr\"obner basis was introduced for shuffle operads with respect to a total order on tree monomials. In this article, we introduce the structure of shuffle polygraphs as a categorical model for rewriting in shuffle operads, which generalizes the Gr\"obner bases approach by removing the constraint of a monomial order for the orientation of the rewriting rules. We define w-operads as internal w-categories in the category of shuffle operads. We show how to extend a convergent shuffle polygraph into a shuffle polygraphic resolution generated by the overlapping branchings of the original polygraph. Finally, we prove that a shuffle operad presented by a quadratic convergent shuffle polygraph is Koszul.
Published: 2020

50. Computing Free Non-commutative Groebner Bases over Z with Singular:Letterplace

Author: Levandovskyy, Viktor, Metzlaff, Tobias, Zeid, Karim, Rheinisch-Westfälische Technische Hochschule Aachen University (RWTH), AlgebRe, geOmetrie, Modelisation et AlgoriTHmes (AROMATH), Inria Sophia Antipolis - Méditerranée (CRISAM), Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-National and Kapodistrian University of Athens (NKUA), European Project: 813211,H2020-EU.1.3. - EXCELLENT SCIENCE - Marie Skłodowska-Curie Actions (Main Programme), H2020-EU.1.3.1. - Fostering new skills by means of excellent initial training of researchers ,10.3030/813211,POEMA(2019), Rheinisch-Westfälische Technische Hochschule Aachen (RWTH), and European Project: 813211,H2020,POEMA(2019)
Subjects: Non-commutative algebra, Mathematics::Commutative Algebra, Rings and Algebras (math.RA), [MATH.MATH-RA]Mathematics [math]/Rings and Algebras [math.RA], FOS: Mathematics, Coefficients in rings, Mathematics - Rings and Algebras, Groebner bases, [MATH]Mathematics [math], Algorithms
Abstract: With this paper we present an extension of our recent ISSAC paper about computations of Groebner(-Shirshov) bases over free associative algebras Z. We present all the needed proofs in details, add a part on the direct treatment of the ring Z/mZ as well as new examples and applications to e.g. Iwahori-Hecke algebras.The extension of Groebner bases concept from polynomial algebras over fields to polynomial rings over rings allows to tackle numerous applications, both of theoretical and of practical importance.Groebner and Groebner-Shirshov bases can be defined for various non-commutative and even non-associative algebraic structures. We study the case of associative rings and aim at free algebras over principal ideal rings. We concentrate ourselves on the case of commutative coefficient rings without zero divisors (i.e. a domain). Even working over Z allows one to do computations, which can be treated as universal for fields of arbitrary characteristic. By using the systematic approach, we revisit the theory and present the algorithms in the implementable form. We show drastic differences in the behavior of Groebner bases between free algebras and algebras, close to commutative.Even the process of the formation of critical pairs has to be reengineered, together with the implementing the criteria for their quick discarding.We present an implementation of algorithms in the Singular subsystem called Letterplace, which internally uses Letterplace techniques (and Letterplace Groebner bases), due to La Scala and Levandovskyy. Interesting examples and applications accompany our presentation.
Published: 2020

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