Invariants des hypermatrices

Abstract : This thesis is devoted to the theory of the invariants of hypermatrices. The origin of the theory of invariants dates from the mid-XIX th century. The general problem, such as it was stated by Cayley in 1843, consists in finding a description of the algebra of the polynomial invariants in the aim of automating the geometric reasoning. Enough quickly strong limitations due to the size of calculations appeared and found this discipline are less and less studied until in the years 1950 when the geometrical theory of the invariants was developed. Nowadays, the progress in the computer sciences allow to complete calculations which had not been able to succeed and to treat new cases. The interest of this discipline increased recently thanks to the discovery of a link with a notion of quantum mechanics which is fundamental in quantum computation: the entanglement. The notion of entanglement is due to Einstein, Podolsky and Rozen who saw in it a proof of the nonconsistency of the quantum theory known under the name of paradox EPR. Since, many experiments, like the experiment of Alain Aspect, show the existence of the entangled states. This document is composed of two parts. In the first, we expose the fundamental techniques of the theory of the invariants and the links with entanglement such as it was proposed by A. Klyachko. We show that the implementation of the Gordan algorithm allows to compute sets of fundamental invariants and covariants of certain multilinear forms. In particular, we illustrate this by giving a complete system of generators for the algebra of the covariants of a quadri-linear form (system of 4-qubits). We show also the limits of this approach: starting the computation for the quinti-linear form (system of 5-qubits), we see that the complexity of the algebras prohibits the generalization of this method. Worse, even if the description of these algebras in term of generators and relations could be obtained, this one would be humanly not usable. We propose then to consider only certain invariants having remarkable properties (for example by studying the structure of Cohen-Macaulay of these algebras). The second part is devoted to a particular invariant : the hyperdeterminant. This polynomial generalizes the determinant in the simplest possible way since it can be defined as a multi-alternated sum on the product of several symmetrical groups. After giving some general properties, we study special cases like the Hankel hyperdeterminants, or the hyperdeterminants of tensors whose entries depend only on the pgcd of indices etc.. Many results of this part are applied to the calculation of iterated integrals In particular, we give a generalization of the Heine theorem an alternative proof of the Selberf integral and generalizations of the de Bruijn integrals.
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Submitted on : Monday, February 11, 2008 - 4:34:44 PM
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Jean-Gabriel Luque. Invariants des hypermatrices. Autre [cs.OH]. Université de Marne la Vallée, 2008. ⟨tel-00250312⟩

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