Two Problems Related to the Circle Method

Abstract

This thesis consists of two parts. In the first part, we consider a system of polynomials with integer coefficients of the same degree with non-singular local zeros and the number of variables large compared to the Birch singular locus of these polynomials. Generalising the work of Birch on the circle method, we find quantitative asymptotics (in terms of the maximal coefficient of these polynomials) for the number of integer zeros of this system within a growing box. Using a quantitative version of the Nullstellensatz, we obtain an upper bound on the smallest integer zero provided the system of polynomials is non-singular.

In the second part we compute a q-hypergeometric generating series for overpartitions of a given rank d where the difference between two successive parts may be odd only if the larger part is overlined. We show that all coefficients are divisible by 3 except for the coefficient of q^(d^2).