  We give upper bounds for the number of integral solutions of bounded height
to a system of equations $f_i(x_1,\ldots,x_n) = 0$, $1 \leq i \leq r$, where
the $f_i$ are polynomials with integer coefficients. The estimates are obtained
by generalising an approach due to Heath-Brown, using a certain $q$-analogue of
van der Corput's method, to the case of systems of polynomials of differing
degree. Our results apply for a wider range of $n$, in terms of the degrees of
the polynomials $f_i$, than bounds obtained with the circle method.
