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|All Authors / Contributors:||
Raphaël Astier; Vincent Cossart; Université de Versailles-Saint-Quentin-en-Yvelines.
|Notes:||Résumé en français.|
|Description:||48 p. : ill. ; 30 cm.|
|Responsibility:||Raphaël Astier ; sous la dir. de Vincent Cossart.|
This thesis deals with uniformization, in characteristic p>0, of a rational valuation, in special cases where this valuation is centered on a singularity locally defined by the following equations :- either zp̂+f(x,y)=0, with f not a p-th power, and ordf >p,- or zp̂+e(x,y)z+f(x,y)=0, with ord (ez+f)>p (Artin-Schreier's case).Historically, it was in such cases that all difficulty of resolving surfaces in positive characteristic was concentrated.The novelty bringed in this work consists first in giving a bound to theminimum number of closed point's blowing-ups needed to uniformize, and second in anticipating (from the first ring) the Newton polygon's evolution and the parameter's choice for the successive blowing-ups along the valuation. In a first part, we come back on the Giraud's normal form of f in O_X(X)where X is a two dimensional regular scheme of characteristic p. The startingpoint is an polynomial expansion of f with a generating sequence for the valuation. We can then study and anticipate the behavior of this expansion and the associated Newton polygon modulo a p-th power. We then give a bound on the maximum number of blowing-ups needed for this polygon to become minimal, with only one vertex, and of maximal height one. This case correspond to the normal form of f.In a second part, using this results for the two above-mentionned cases, wegive an algorithm witch anticipate, in the first ring, the translations on zneeded to keep a minimal Newton polygon during the blowing-ups sequence (alongthe valuation), and we quantify the maximal size of such a sequence with last ring corresponding to a quasi-ordinary singularity.