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.Comment: Accepted for publication in Acta Arithmetica. Added a few pages so
that familiarity with Birch's work is no longer assumed; 24 page