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

Abstract

Let $F \in \mathbb{Z}[x_1, \ldots, x_n]$ be a homogeneous form of degree $d \geq 2$, and let $V_F^*$ denote the singular locus of the affine variety $V(F) = \{ \mathbf{z} \in {\mathbb{A}}^n_{\mathbb{C}}: F(\mathbf{z}) = 0 \}$. In this paper, we prove the existence of integer solutions with prime coordinates to the equation $F(x_1, \ldots, x_n) = 0$ provided $F$ satisfies suitable local conditions and $n - \dim V_F^* \geq 7 d (2d-1) 4^d + 4 (d-1) (12d - 1) 2^d + 12d$. The result is obtained by using the identity $\Lambda = \mu * \log$ for the von Mangoldt function and optimizing various parts of the argument in the author's previous work, which made use of the Vaughan identity and required $n - \dim V_F^* \geq 2^8 3^4 5^2 d^3 (2d-1)^2 4^{d}$.