Random Diophantine Equations in the Primes II

Let $d\ge 2$ and $n\ge d$ with $(d,n)\mathrm{\notin }\{(2,2),(3,3)\}$. 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).