  Let $F_1,\ldots,F_R$ be homogeneous polynomials of degree $d\ge 2$ with
integer coefficients in $n$ variables, and let $\mathbf{F}=(F_1,\ldots,F_R)$.
Suppose that $F_1,\ldots,F_R$ is a non-singular system and $n\ge
4^{d+2}d^2R^5$. We prove that there are infinitely many solutions to
$\mathbf{F}(\mathbf{x})=\mathbf{0}$ in prime coordinates if (i)
$\mathbf{F}(\mathbf{x})=\mathbf{0}$ has a non-singular solution over the
$p$-adic units $\U_p$ for all prime numbers $p$, and (ii)
$\mathbf{F}(\mathbf{x})=\mathbf{0}$ has a non-singular solution in the open
cube $(0,1)^n$.
