Les singularités des polynômes à l'infini et les compactifications toriques

Abstract : This thesis is devoted to the study of topology of complex polynomials. In the preliminaries, we present the various techniques we used, like stratified vector field and control conditions about this vector field, and toric varieties. We also introduce preparatories results about properties of toric compactification of polynomial's fibers.
In chapter 2, we give main results in the case of weighted toric compactification of affine space C^n. We prove affine polynomial triviality with the help of tame hypothesis on Malgrange-Paunescu's weight gradient : |grad_Wf(z)|_W is lower bounded. Thanks to this hypothesis we also prove that Kuo-Paunescu vector field after toric modification become a control vector field in relation to the divisor at infinity. This last condition give us the main condition : non-characteristic condition. We deduce local triviality in a divisor point.

Chapter 3 is based on Hamm, Lê and Mebkhout works. It describe connection between the non-characteristic condition obtain in chapter 2 and the notion of vanishing cycles and also local triviality.

Chapter 4 generalised theorem of chapter 2 for any toric compactification of affine space C^n.
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David Alessandrini. Les singularités des polynômes à l'infini et les compactifications toriques. Mathématiques [math]. Université d'Angers, 2002. Français. ⟨tel-00002671v3⟩

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