# Recent results in the theory of constant reductions

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Recent results in the theory of constant reductions JOURNAL DE THÉORIE DES NOMBRES DE BORDEAUX BARRYGREEN Recent results in the theory of constant reductions Journal de Théorie des Nombres de Bordeaux, tome 3, no 2 (1991), p. 275- 310. <http://www.numdam.org/item?id=JTNB_1991__3_2_275_0> © Université Bordeaux 1, 1991, tous droits réservés. L’accès aux archives de la revue « Journal de Théorie des Nombres de Bordeaux » (http://jtnb.cedram.org/) implique l’accord avec les condi- tions générales d’utilisation (http://www.numdam.org/legal.php). Toute uti- lisation commerciale ou impression systématique est constitutive d’une infraction pénale. Toute copie ou impression de ce fichier doit conte- nir la présente mention de copyright. Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques http://www.numdam.org/ , 275-3 Recent results in the theory of constant reductions by Barry GREEN The aim of this paper is to give a survey of recent results in the theory of constant reductions and in particular to examine the way in which the approaches via rigid analytic geometry or alternatively function field theory have been used to prove these results. From both these areas there has been an interaction of ideas and approaches to solving problems and in this paper we have attempted to illustrate this by the results we discuss. These results are all of algebraic or geometric nature and in some cases special forms were already known from function field theory, rigid analytic geometry, algebraic geometry and valuation theory. We also illustrate how the model theory of valued function fields is used to provide the framework in which unknown cases or more general forms of these results ca.n be proved. Examples of this phenomenon which are discussed in this paper are: - a theorem on the existence of regular functions for valued function fields; - a stable reduction theorem for curves over an arbitrary valuation ring with algebraically closed quot

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