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

TL;DR

This paper proves the existence of prime solutions to certain homogeneous Diophantine equations under specific geometric and local conditions, significantly improving previous bounds on the number of variables needed.

Contribution

It establishes new conditions ensuring prime solutions for homogeneous forms, reducing the variable count requirement from exponential tower bounds to a polynomial bound.

Findings

Prime solutions exist under new geometric and local conditions.

Improved bounds on the number of variables needed for solutions.

Extension of previous results to broader classes of forms.

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 $V(F) = \{ \mathbf{z} \in {\mathbb{C}}^n: F(\mathbf{z}) = 0 \}$. In this paper, we prove the existence of integer solutions with prime coordinates to the equation $F(x_1, \ldots, x_n) = 0$ provided $F$ satisfies suitable local conditions and $n - \dim V_F^* \geq 2^8 3^4 5^2 d^3 (2d-1)^2 4^{d}$. 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 $n - \dim V_F^*$ to be an exponential tower in $d$.