Extensiones del problema de Büchi a distintas estructuras y potencias más altas.
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Date
2010
Authors
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Journal ISSN
Volume Title
Publisher
Universidad de Concepción.
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
Keywords
Campos Finitos (Álgebra), Ecuaciones Diferenciales, Büchi, Problema