  Let $F_1,\dotsc,F_R$ be quadratic forms with integer coefficients in $n$
variables. When $n\geq 9R$ and the variety $V(F_1,\dotsc,F_R)$ is a smooth
complete intersection, we prove an asymptotic formula for the number of integer
points in an expanding box at which these forms simultaneously vanish, which in
particular implies the Hasse principle for $V(F_1,\dotsc,F_R)$. Previous work
in this direction required $n$ to grow at least quadratically with $R$. We give
a similar result for $R$ forms of degree $d$, conditional on an upper bound for
the number of solutions to an auxiliary inequality. In principle this result
may apply as soon as $n> d2^dR$. In the case that $d\geq 3$, several strategies
are available to prove the necessary upper bound for the auxiliary inequality.
In a forthcoming paper we use these ideas to apply the circle method to
nonsingular systems of forms with real coefficients.
