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

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 $V(F)=\{\mathbf{z}\in {\mathbb{A}}_{\mathbb{C}}^n:F(\mathbf{z})=0\}$. 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 $n-\dim V_F^{\ast }\ge 7d(2d-1)4^d+4(d-1)(12d-1)2^d+12d$. The result is obtained by using the identity $\Lambda =\mu \ast \log$ 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 $n-\dim V_F^{\ast }\ge 2^8{3}^4{5}^2{d}^3(2d-1{)}^2{4}^d$.

42 pages. arXiv admin note: text overlap with arXiv:2105.12435