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