Potencias en subsucesiones de progresiones aritméticas en anillos de funciones y un problema de indecidibilidad.
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Date
2011
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Publisher
Universidad de Concepción
Abstract
This thesis consists of two main theorems (Theorems 2 and 6) on the arithmetic of polynomials and some consequences in Mathematical Logic.
A simple corollary of our first main theorem, inspired by a paper on powers
in arithmetic progressions by Hajdu [10], states in particular that given
any field F the quantity a!+b, where a and b are coprime polynomials in F[t], cannot be a power in F[t] for more than M = 4 distinct values of ! ! F, unless both a and b have zero derivative.
Here by power, we mean a k-th power for some k " 2. This corollary was
actually our very first result and inspired the rest of the thesis. On the one
hand, one could naturally try to generalize it in the following ways: 1. Is the condition on coprimality really necessary? If not, how small should the degree of the greatest common divisor of a and b be in order to ensure the existence of an M (guessing that the more a and b have factors in common, the bigger should be M); 2. What about other rings of functions? (such that subrings of function fields, rings of analytic or meromorphic functions, . . . ) 3. What about considering expressions of the form a!2 + b! + c instead of a! + b? What about higher powers? In the work presented here, we deal with generalizations of type 1 (Theorem 2) and of type 3 (Theorem 6).
On the other hand, the statement above and its possible generalizations
should say something about the first order language LP = {0, 1,+, P} (or
languages containing LP ), where P is a unary relation symbol and P(x) is
interpreted as “x is a power”.
Though to the best of our knowledge this language has not been previously
studied by logicians (though Macintyre’s language [12] is somewhat
related to it, as it contains a predicate Pk for each k " 2 interpreted as
“Pk(x) if and only if x is a k-th power”), the study of consecutive powers
and powers in arithmetic progression over the integers has a long history.
In 1640, Fermat conjectured that there does not exist four squares in arithmetic progression, and this was proved later on by Euler.
Description
Tesis presentada para optar al grado de Magíster en Matemática.
Keywords
Anillos (Álgebra), Problemas Ejercicios etc., Potencias., Aritmética, Problemas Ejercicios Etc.