Random Diophantine Equations in the Primes II

TL;DR

This paper investigates the solvability of homogeneous Diophantine equations in prime numbers, demonstrating a local-global principle for most cases by adapting recent advanced methods.

Contribution

It extends previous work by establishing a local-global principle for prime solutions to a broad class of Diophantine equations using modern techniques.

Findings

A local-global principle holds for almost all such equations.

The methods of Browning, Le Boudec, and Sawin are successfully adapted.

The results generalize prior work on Diophantine equations in primes.

Abstract

Let $d\ge 2$ and $n\ge d$ with $(d,n)\notin \{(2,2),(3,3)\}$. We consider homogeneous Diophantine equations of degree $d$ in $n+1$ variables and whether they have solutions in the primes. In particular, we show that a certain local-global principle holds for almost all such equations, following on from previous work of the author arXiv:2305.06306. We do this by adapting the methods of Browning, Le Boudec and Sawin (Annals, 2023).