Meditatio algebraica noncommutativa : Newton-Girard and Waring-Lagrange theorems for two non-commuting variables
Analyse Fonctionnelle
In 1629 Albert Girard, a French Huguenot exiled to the Netherlands, gave formulae for the
power sums of several commuting variables in terms of the elementary symmetric functions;
his result was subsequently often attributed to Newton.
Over a century later Waring proved in his Meditationes algebraicae
that an arbitrary symmetric polynomial in finitely many commuting variables
could be expressed as a polynomial in the elementary symmetric functions of those variables.
In 1939 Margarete Wolf studied the analogous questions for non-commuting variables. She showed that there is no finite algebraic basis for the algebra of symmetric functions in $d > 1$ non-commuting variables, so there is no finite set of `elementary symmetric functions' in the non-commutative case.
Nevertheless, Jim Agler, John McCarthy and I have proved analogues of Girard's and Waring's
theorems for symmetric functions in two non-commuting variables. We find three free polynomials $f, g, h$ in two non-commuting indeterminates $x, y$ such that every symmetric polynomial in $x$ and $y$ can be written as a polynomial in $f, g, h$ and $1/g$. In particular, power sums can be written explicitly in terms of $f,g$ and $h$. To do this we developed the notion of a non-commutative manifold.
- Accueil
- Annuaire
- Equipes
- Evènements
- Congrès
- Invités
- Séminaires, Groupes de Travail et Colloquium
- Séminaires
- Analyse Complexe et Equations Différentielles
- Analyse Fonctionnelle
- Analyse Numérique et Equations Aux Dérivées Partielles
- Arithmétique
- Formes Automorphes
- Géométrie Algébrique
- Géométrie des espaces singuliers
- Géométrie Dynamique
- Histoire des Mathématiques
- Physique Mathématique
- Probabilités et Statistique
- Singularités et Applications
- Théorie Analytique et Analyse Harmonique
- Topologie
- Colloquium
- Groupes de Travail
- Analyse harmonique et théorie analytique
- Autour des fractales
- Calcul de Malliavin et processus fractionnaires
- Déformations des singularités de surfaces
- Equations aux dérivées partielles
- Extraction du signal
- Fondements mathématiques du deep learning
- Géométrie Non-Archimédienne
- Géométrie Stochastique
- Idéaux de Hodge
- Leçons d'Analyse
- Matrices Aléatoires
- Probabilités
- Statistique et Grande Dimension
- Systèmes Dynamiques
- Topologie
- W-algèbres
- Doctorants et Post-doctorants
- Séminaires
- Soutenances
- Anciens Séminaires et Groupes de Travail
- Formation par la Recherche
- Laboratoire
- Liens utiles
- Projets
- Recrutements
- Services