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

Let $F\in \mathbb{Z}[x_1,\dots ,x_n]$ be a homogeneous form of degree $d \geq 2$,andlet$V_F^*$denotethesingularlocusoftheaffinevariety$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 $F$ 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 (B. Cook and A. Magyar, `Diophantine equations in the primes’. Invent. Math. 198 (2014), 701-737), which required $n-\dim V_F^{\ast }$ to be an exponential tower in $d$.