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Title: Extensiones del problema de Büchi a distintas estructuras y potencias más altas.
Authors: Vidaux Negre, Xavier, supervisor de grado
Pastén Vásquez, Héctor Hardy
Keywords: Campos Finitos (Álgebra);Ecuaciones Diferenciales;Büchi, Problema
Issue Date: 2010
Publisher: Universidad de Concepción, Facultad de Ciencias Físicas y Matemáticas, Departamento de Matemática.
Abstract: In any commutative ring A with unit, Büchi sequences are those sequences whose second di erence of squares is the constant sequence (2). Sequences of elements xn satisfying x2 n = (x + n)2 for some xed x are Büchi sequences that we call trivial. Since we want to study sequences whose elements do not belong to certain subrings (e.g. for elds of rational functions F(z) over a eld F we are interested in sequences that are not over F) the concept of trivial sequences may vary. Büchi's Problem for a ring A asks whether there exists a positive integer M such that any Büchi sequence of length M or more is trivial. We survey the current status of knowledge for Büchi's problem and its analogues for higher-order di erences and higher powers. We propose several new and old open problems. We present a few new results and various sketches of proofs of old results (in particular : Vojta's conditional proof for the case of integers and a quite detailed proof for the case of polynomial rings in characteristic zero), and present a new and short proof of the positive answer to Büchi's problem over nite elds with p elements (originally proved by Hensley). We discuss applications to Logic (which were the initial aim for solving these problems).
Description: Tesis para optar al grado de Magíster en Matemática
Appears in Collections:Ciencias Físicas y Matemáticas - Tesis Magister

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