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La présente étude décrit les conditions et les modalités concrètes de différents types d’exposés en mathématiques : de l’exposé écrit pour une publication aux diverses formes d’exposition orale, qu’elles soient de vulgarisation, ou qu’elles s’adressent à un public général de mathématiciens ou restreint à la petite communauté de spécialistes d’un champ de recherche particulier. Elle montre combien la persuasion mise en oeuvre lors de l’exposé d’un résultat renvoie, pour l’orateur ou pour le lecteur, à la possibilité d’en rejouer la démonstration, soit de rester en position active vis-à-vis de l’énonciation., This analysis describes the conditions and the actual modalities of different types of displays in mathematics: written displays in published articles as well as oral presentations directed to a public with some or no specific knowledge in the field, or an address given within a small circle of experts in a specific field of research. The article shows how the persuasiveness of the presentation, for orator and reader alike, depends on being able to replay the demonstration, to remain active to what’s being said.

— Let (K, ν) be a valued field and L a finite cyclic extension of K defined by L = K[x]/(P ), then any valuation of L which extends ν defines a pseudo-valuation ζ onK[x] whose kernel is the principal ideal (P ). We know how to associate to ζ a family of valuations on K[x], called an admissible family, which is explicitely constructed by augmented valuations and limit augmented valuations. We give a necessary and sufficient condition for a valuation of K[x] to belong to an admissible family associated to a pseudo-valuation ζ which corresponds to a valuation of L, this condition depends only on the polynomial P . On the way we can determine all the valuations of L which extend the valuation ν of K. To give this condition we define the Newton polygon associated to P , to a polynomial φ and to a valuation μ of K[x].