  Let $\mathbf{f} = (f_1, \ldots, f_R)$ be a system of polynomials with integer
coefficients in which the degrees need not all be the same. We provide
sufficient conditions for which the system of equations $f_j (x_1, \ldots, x_n)
= 0 \ (1 \leq j \leq R)$ satisfies a general local to global type statement,
and has a solution where each coordinate is prime. In fact we obtain the
asymptotic formula for number of such solutions, counted with a logarithmic
weight, under these conditions. We prove the statement via the Hardy-Littlewood
circle method. This is a generalization of the work of B. Cook and \'{A}.
Magyar, where they obtained the result when the polynomials of $\mathbf{f}$ all
have the same degree. Hitherto, results of this type for systems of polynomial
equations involving different degrees have been restricted to the diagonal
case.
