Schémas de Hilbert et décompositions de tenseurs

Jerome Brachat 1
1 GALAAD - Geometry, algebra, algorithms
CRISAM - Inria Sophia Antipolis - Méditerranée , UNS - Université Nice Sophia Antipolis, CNRS - Centre National de la Recherche Scientifique : UMR6621
Abstract : This thesis consists of two parts. The first one contains chapter 2 and 3 and is about the Hilbert scheme. These chapters correspond to joint works with M.E. Alonso and B. Mourrain : [3] and with P. Lella, B. Mourrain and M. Roggero : [10]. We are interested in the equations that define it as a closed sub-scheme of the Grassman- nian and especially their degree. We will give new global equations, more simple than those already known. Chapter 2 is about the case of constant Hilbert polynomial equal to μ. First, we will briefly recall the defini- μ tions and propositions related to the Hilbert functor associated to μ, denoted by HilbPn . Then we will prove that it is representable, we will use a local approach and build a covering of open representable sub-functors whose equations correspond to commutation relations that characterize border basis. The scheme that repre- The second part of this thesis is concerned with tensors decomposition, chapter 4. We will begin with the sym- metric case which corresponds to a joint work with P. Comon, B. Mourrain and E. Tsigaridas : [9]. We will extend the algorithm devised by Sylvester for the binary case. We will use a dual approach and give necessary and sufficient conditions for the existence of a decomposition of a given rank, using Hankel operators. We will deduce an algorithm for the symmetric case. Finally, we will conclude by studying the case of general tensors which corresponds to a joint work with A. Bernardi, P. Comon and B. Mourrain : [6]. In particular, we will prove how the formalism that as been used so far for the symmetric case, can be extended to solve the problem. μμn sents HilbPn is called the Hilbert scheme associated to μ and is denoted by Hilb (P ). Then, thanks to the theorems of Persistence and Regularity of Gotzmann, we will give a global description of Hilbμ(Pn). We will provide a set of homogeneous equations of degree 2 in the Plücker coordinates that characterizes Hilbμ(Pn) as a closed sub-scheme of the Grassmannian. We will finally conclude this chapter by studying the tangent plan of the Hilbert scheme. Chapter 3 deals with the general case of Hilbert scheme associated to a Hilbert polynomial P of degree d ≥ 0, denoted by HilbP (Pn). We will generalize chapter 2, giving homogeneous equations of degree d + 2 in the Plücker coordinates. The second part of this thesis is concerned with tensors decomposition, chapter 4. We will begin with the symmetric case which corresponds to a joint work with P. Comon, B. Mourrain and E. Tsigaridas : [9]. We will extend the algorithm devised by Sylvester for the binary case. We will use a dual approach and give necessary and sufficient conditions for the existence of a decomposition of a given rank, using Hankel operators. We will deduce an algorithm for the symmetric case. Finally, we will conclude by studying the case of general tensors which corresponds to a joint work with A. Bernardi, P. Comon and B. Mourrain : [6]. In particular, we will prove how the formalism that as been used so far for the symmetric case, can be extended to solve the problem.
Keywords : Tensor
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Theses
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Jerome Brachat. Schémas de Hilbert et décompositions de tenseurs. Mathématiques [math]. Université Nice Sophia Antipolis, 2011. Français. ⟨tel-00620047⟩

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