Vidaux Negre, XavierPastén Vásquez, Héctor Hardy2021-05-182024-05-152024-08-282021-05-182024-05-152024-08-282010https://repositorio.udec.cl/handle/11594/5877Tesis para optar al grado de Magíster en MatemáticaIn 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).engCreative Commoms CC BY NC ND 4.0 internacional (Atribución-NoComercial-SinDerivadas 4.0 Internacional)Campos Finitos (Álgebra)Ecuaciones DiferencialesBüchi, ProblemaTesis