Extensions de valuation et polygone de Newton. (Extensions of a valuation and Newton polygon.) (French. English summary) Zbl 1170.13003

Vaquié, Michel

Ann. Inst. Fourier 58, No. 7, 2503-2541 (2008).

Let \((K,\nu )\) be a valued field and \(L=K(\theta)\) a finite cyclic extension of \(K\). The aim of this paper is to determine all valuations extending \(\nu\). \(L\) is isomorphic to \(K[x]/(P)\), and any valuation of \(L\) which extends \(\nu \) defines a pseudo-valuation \(\zeta \) on \(K[x]\) whose kernel is the principal ideal \((P)\). The author associates to \(\zeta \) a family of valuations on \(K[x]\), called an admissible family, which is explicitly constructed by augmented valuations and limit augmented valuations. The author gives a necessary and sufficient condition for a valuation of \(K[x]\) to belong to an admissible family associated to a pseudo-valuation \(\zeta \) which corresponds to a valuation of \(L\), this condition depends only on the polynomial \(P\). The author determines all the valuations of \(L\) which extend the valuation \(\nu \) of \(K\). In order to solve this problem the author defines the Newton polygon \({\mathcal PN}(P;\varphi;\mu)\), associated to \(P\), to a polynomial \(\varphi \) and to a valuation \(\mu \) of \(K[x]\).

Reviewer: Marcel Morales (Saint-Martin-d’Heres)

MSC:

13A18	Valuations and their generalizations for commutative rings
12J10	Valued fields
14E15	Global theory and resolution of singularities (algebro-geometric aspects)

Keywords:

valuation; extension; Newton polygon

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