Diophantine equations in primes: Density of prime points on affine hypersurfaces

Let F ∈ Z[x1, . . . , xn] be a homogeneous form of degree d ≥ 2, and let V ∗ F denote the singular locus of the affine variety V (F ) = {z ∈ C : F (z) = 0}. In this paper, we prove the existence of integer solutions with prime coordinates to the equation F (x1, . . . , xn) = 0 provided F satisfies suitable local conditions and n − dimV ∗ F ≥ 235d(2d− 1)4. Our result improves on what was known previously due to Cook and Magyar (B. Cook and Á. Magyar, ‘Diophantine equations in the primes’. Invent. Math. 198 (2014), 701-737), which required n− dimV ∗ F to be an exponential tower in d.