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

TL;DR

This paper establishes the existence of prime solutions to certain homogeneous Diophantine equations under specific geometric and local conditions, improving previous bounds through refined analytic techniques.

Contribution

It proves prime solutions for homogeneous forms with fewer variables than previously required by optimizing analytic methods and leveraging the von Mangoldt function identity.

Findings

Existence of prime solutions under new variable bounds

Improved bounds compared to previous work

Application of von Mangoldt function identity in Diophantine analysis

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 $n - \dim V_F^* \geq 2^8 3^4 5^2 d^3 (2d-1)^2 4^{d}$.