A local-global principle holds for prime solutions of almost all homogeneous Diophantine equations of degree d in n+1 variables (d≥2, n≥d, excluding (2,2) and (3,3)).
Yamagashi, Diophantine equations in primes: Density of prime points on affine hypersurfaces, Duke Mathematical Journal 171 (2022), 831-884
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Random Diophantine Equations in the Primes II
A local-global principle holds for prime solutions of almost all homogeneous Diophantine equations of degree d in n+1 variables (d≥2, n≥d, excluding (2,2) and (3,3)).