Zeroes of polynomials in many variables with prime inputs

In this paper we apply the Hardy-Littlewood circle method to 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 and does not require geometric data. This extends the work of B. Cook and A. Magyar for hypersurfaces of degree $2$ and $3$.

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