  We consider systems $\vec{F}(\vec{x})$ of $R$ homogeneous forms of the same
degree $d$ in $n$ variables with integral coefficients. If $n\geq d2^dR+R$ and
the coefficients of $\vec{F}$ lie in an explicit Zariski open set, we give a
nonsingular Hasse principle for the equation $\vec{F}(\vec{x})=\vec{0}$,
together with an asymptotic formula for the number of solutions to in integers
of bounded height. This improves on the number of variables needed in previous
results for general systems $\vec{F}$ as soon as the number of equations $R$ is
at least 2 and the degree $d$ is at least 4.
