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Title: | Potencias en subsucesiones de progresiones aritméticas en anillos de funciones y un problema de indecidibilidad. |

Other Titles: | Powers in subsequences of arithmetic progressions in rings of functions and a problem of undecidability. |

Authors: | Vidaux, Xavier, supervisor de grado García Fritz, Natalia Cristina |

Keywords: | Anillos (Álgebra) -- Problemas, Ejercicios, etc.;Potencias.;Aritmética -- Problemas, Ejercicios, Etc. |

Issue Date: | 2011 |

Publisher: | Universidad de Concepción, Facultad de Ciencias Físicas y Matemáticas, Departamento de Matemática. |

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 para optar al grado de Magíster en Matemática. |

URI: | http://repositorio.udec.cl/jspui/handle/11594/5849 |

metadata.dc.source.uri: | http://ezpbibliotecas.udec.cl/login?url=http://tesisencap.udec.cl/concepcion/garcia_f_n/index.html |

Appears in Collections: | Ciencias Físicas y Matemáticas - Tesis Magister |

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