  We consider a system of $R$ cubic forms in $n$ variables, with integer
coefficients, which define a smooth complete intersection in projective space.
Provided $n\geq 25R$, we prove an asymptotic formula for the number of integer
points in an expanding box at which these forms simultaneously vanish. In
particular we can handle systems of forms in $O(R)$ variables, previous work
having required that $n \gg R^2$. One conjectures that $n \geq 6R+1$ should be
sufficient. We reduce the problem to an upper bound for the number of solutions
to a certain auxiliary inequality. To prove this bound we adapt a method of
Davenport.
