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 VF∗ denote the singular locus of the affine variety V(F)={z∈Cn:F(z)=0}. In this paper, we prove the existence of integer solutions with prime coordinates to the equation F(x1,…,xn)=0 provided that F satisfies suitable local conditions and n−dimVF∗≥283452d3(2d−1)24d. Our result improves on what was known previously due to Cook and Magyar, which required n−dimVF∗ to be an exponential tower in d.