1. Extensions de valuation et polygone de Newton
- Author
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Michel Vaquié, Institut de Mathématiques de Toulouse UMR5219 (IMT), Institut National des Sciences Appliquées - Toulouse (INSA Toulouse), Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-Université Toulouse 1 Capitole (UT1), Université Fédérale Toulouse Midi-Pyrénées-Université Fédérale Toulouse Midi-Pyrénées-Université Toulouse - Jean Jaurès (UT2J)-Université Toulouse III - Paul Sabatier (UT3), Université Fédérale Toulouse Midi-Pyrénées-Centre National de la Recherche Scientifique (CNRS), Université Toulouse Capitole (UT Capitole), Université de Toulouse (UT)-Université de Toulouse (UT)-Institut National des Sciences Appliquées - Toulouse (INSA Toulouse), Institut National des Sciences Appliquées (INSA)-Université de Toulouse (UT)-Institut National des Sciences Appliquées (INSA)-Université Toulouse - Jean Jaurès (UT2J), Université de Toulouse (UT)-Université Toulouse III - Paul Sabatier (UT3), and Université de Toulouse (UT)-Centre National de la Recherche Scientifique (CNRS)
- Subjects
Polynomial ,Algebra and Number Theory ,13A18 (12J10 14E15) ,010102 general mathematics ,Mathematical analysis ,Field (mathematics) ,Newton polygon ,Extension (predicate logic) ,01 natural sciences ,Combinatorics ,Kernel (algebra) ,Principal ideal ,0103 physical sciences ,010307 mathematical physics ,Geometry and Topology ,Limit (mathematics) ,0101 mathematics ,ComputingMilieux_MISCELLANEOUS ,Valuation (finance) ,Mathematics - Abstract
— 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].
- Published
- 2008
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