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Indépendance algébrique de logarithmes en caractéristique P

Published online by Cambridge University Press:  17 April 2009

Laurent Denis
Laurent Denis
Affiliation:
Laboratoire Paul Painlevé UMR CNRS 8524, U.F.R. de Mathématiques Pures et Appliquées, Bât. M2, Université des Sciences et Technologies de Lille 1, 59665 Villeneuve d'Ascq Cedex, France, e-mail: [email protected]
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Let k be the rational function field over the field with q elements with characteristic p. Since the work of Carlitz we know in this situation the function ζ analog of the Riemann zeta function and the function Logφ analog of the usual logarithm. We will show two main results. Firstly, if ξ denotes the fundamental period of Carlitz module, we prove that ξ, ζ(1),…, ζ(p – 2) are algebraically independent over k. Secondly if α1,…, αn are rational elements (of degree less than q/(q − 1) to ensure convergence of the logarithm) such that Logφ α1,…, Logφ αn are linearly independent over k then they are algebraically independent over k. The point is to find suitable functions taking these values and for which Mahler's method can be used.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 74 , Issue 3 , December 2006 , pp. 461 - 470
Copyright
Copyright © Australian Mathematical Society 2006

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