Diophantine equations of binomial coefficients and exponential sums of symmetric Boolean functions

Abstract

In this thesis, we study the link that exists between solutions to Diophantine equations that involve binomial coefficient over a bounded set of integers and exponential sums of perturbations of symmetric Boolean functions. This link was established by Castro and Medina in [3]. They extended the concepts of trivially balanced functions and sporadic balanced functions to these perturbations. This problem also is similar to the interesting problem of bisecting binomials which was first studied by Ionascu, Stanica and Martinsen [11], but our study is from the point of view of the theory of exponential sums of symmetric Boolean functions.<br /> <br /> Here in this thesis it is presented an identity of two exponential sums of perturbations of two different symmetric Boolean functions. We also study the balancedness of these perturbations of fixed degree when the number of variables grows and we show that these balanced perturbation of fixed degree do not exist when the number of variables grow based on an observation of Canteaut and Videau. Finally, we present some examples of sporadic balanced perturbations and their corresponding Diophantine equation with binomial coefficients.