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Let $\mathbf{f} = (f_1, \ldots, f_R)$ be a system of polynomials with integercoefficients in which the degrees need not all be the same. We providesufficient 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 theasymptotic formula for number of such solutions, counted with a logarithmicweight, under these conditions. We prove the statement via the Hardy-Littlewoodcircle method. This is a generalization of the work of B. Cook and \'{A}.Magyar, where they obtained the result when the polynomials of $\mathbf{f}$ allhave the same degree. Hitherto, results of this type for systems of polynomialequations involving different degrees have been restricted to the diagonalcase.

Prime solutions to polynomial equations in many variables and differing degrees