# La-dimension-deux.

Titre : Décompositions des produits tensoriels déformés dans les espaces de dimension deux.

Auteur : Thierry PERIAT.

Immatriculation : ISBN 978-2-36923-103-5, EAN 9782369231035.

Langue : FR.

Version : 4.

Date de parution : 2 mai 2022.

Nombre de pages : 66.

Document : Methext2d livre 20220502 (584.56 Ko).

Commentaires :

Données élémentaires concernant le traitement de la question (E) en dimension deux :

- Discriminant stratégique et interprétation de ses coefficients.

- Exposé de la méthode extrinsèque en détail : principe, résultats, domaine de validité, analyse logique, avantages et inconvénients, remèdes.

- lien avec les coniques.

- énoncé des solutions.

- Application à la relation de Gauss apparue avec son théorème remarquable.

Retour vers la tête de chapitre consacrée aux : "Méthodes mathématiques".

GB/USA

The most elementary manner to present the so-called (E) question is to relate it to the concept of division. The one-dimensional formulation consists in a simple question asking what the result is when one divides y by x? In all cases, that mental maneuver gives a pair (p, z) which may be generically labeled as (main part, residual part).

A less simple way to introduce that question is to remark that it may be resituated inside the discussion on modular expressions. This is indirectly connecting it to the Langland program.

The complicate formulation of the same question consists of asking it in a (D)-dimensional space, or equivalently consists of trying to realize several divisions at the same time in manipulating D (D-1)-dimensional hyperplanes.

The document, which is exposed here, begins with an analysis of Gauss celebrated remarkable theorem (1827) because it offers a natural opportunity to introduce deformed tensor products acting in a 2-dimensional space. It turns out that that theorem imposes to work with three such products and that the sum of these products is decomposed into three parts in the geodesic basis; each component appearing in that decomposition is a (2-2) matrix.

On the other hand, the same sum can eventually be decomposed in a different way when one is asking the (E) question for each deformed product which is involved in that sum.

This fact is an invitation for deepening the search for general solutions to the question. It is exactly what I start in the fourth chapter; first in specializing the discussion on antisymmetric (2-2-2) cubes, then in considering any (2-2-2) cube.

The extrinsic method comes here to my rescue, and I propose two sub-methods allowing the discovery of solutions. The crucial point to reach the goal lies in the fact that that method forces considering at least two polynomials, one depending on the projectile and another one depending on the target (see my semantic), and that these polynomials may be interdependent.

This work is not yet achieved. You may contribute if you want!