Systems of forms in many variables

We consider systems $vec{F}(vec{x})$ of $R$ homogeneous forms of the same degree $d$ in $n$ variables with integral coefficients. If $ngeq d2^dR+R$ 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.