Forms in prime variables and differing degrees

Let F₁,…,FR be homogeneous polynomials with integer coefficients in n variables with differing degrees. Write F=(F₁,…,FR) with D being the maximal degree. Suppose that F is a nonsingular system and n≥ D² 4D+6R⁵. We prove an asymptotic formula for the number of prime solutions to F(x)=0, whose main term is positive if (i) F(x)=0 has a nonsingular solution over the p-adic units Uₚ for all primes p, and (ii) F(x)=0 has a nonsingular solution in the open cube (0,1)ⁿ. This can be viewed as a smooth local-global principle for F(x)=0 with differing degrees. It follows that, under (i) and (ii), the set of prime solutions to F(x)=0 is Zariski dense in the set of its solutions.