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

Abstract

Let $F\in \mathbb{Z}[x_1,\dots ,x_n]$ be a homogeneous form of degree $d\ge 2$, and let $V_F^{\ast }$ denote the singular locus of the affine variety $V(F)=\{\mathbf{z}\in {\mathbb{C}}^n:F(\mathbf{z})=0\}$. In this paper, we prove the existence of integer solutions with prime coordinates to the equation $F(x_1,\dots ,x_n)=0$ provided that F satisfies suitable local conditions and $n-dim{V}_F^{\ast }\ge 2^8{3}^4{5}^2{d}^3{(2d-1)}^2{4}^d$. Our result improves on what was known previously due to Cook and Magyar, which required $n-dim{V}_F^{\ast }$ to be an exponential tower in d.