On Forms in Prime Variables

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