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Genre et genre residuel des corps de fonctions values

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Abstract

Let L be a function field of one variable over a valued field (K,|.|), and (|.|i), 1⩽i⩽s, be distinct absolute values over L extending |.| such that the residue fields ¯Li are function fields of one variable over the residue field ¯K of (K,|.|). We define the defect of the valued function fields (L,|.|i)/(K,|.|) and prove an inequality between the genus of L/K and that's of ¯Li/¯K which takes into account the defect, the ramification index of (L,|.|i)/(K,|.|) and the constant field of Li/¯K. Our inequality is better than Mathieu's inequality in discretely valued case.

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  1. UER. de Mathématiques et d' Informatique de l'Université de Bordeaux I U.A. 040 226, 351, cours de la Libération, 33405, Talence Cédex, France

    Michel Matignon

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  1. Michel Matignon
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Matignon, M. Genre et genre residuel des corps de fonctions values. Manuscripta Math 58, 179–214 (1987). https://doi.org/10.1007/BF01169090

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