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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 4^D+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.
Liu et al. (Fri,) studied this question.