  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 $3$.
