Title: Quantitative Results on Diophantine Equations in Many Variables,
Abstract: We consider a system of polynomials $f_1,\ldots, f_R\in
\mathbb{Z}[x_1,\ldots, x_n]$ 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.