Zeroes of polynomials with prime inputs and Schmidt's $h$-invariant

In this paper we show that a polynomial equation admits infinitely many prime-tuple solutions assuming only that the equation satisfies suitable local conditions and the polynomial is sufficiently non-degenerate algebraically. Our notion of algebraic non-degeneracy is related to the $h$-invariant introduced by W. M. Schmidt. Our results prove a conjecture of B. Cook and '{A}. Magyar \cite{CM} for hypersurfaces of degrees $2$ and $3$.