Quantitative Results on Diophantine Equations in Many Variables

We consider a system of integer polynomials of the same degree with non-singular local zeros and in many variables. Generalising the work of Birch (1962) we find quantitative asymptotics (in terms of the maximum of the absolute value of the coefficients 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 a quantitative strong approximation result, i.e. an upper bound on the smallest integer zero provided the system of polynomials is non-singular.