PRIME SOLUTIONS TO POLYNOMIAL EQUATIONS IN MANY VARIABLES AND DIFFERING DEGREES

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



$f_{j}(x_{1},\ldots ,x_{n})=0~(1\leqslant j\leqslant R)$


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 Cook and Magyar [‘Diophantine equations in the primes’,
Invent. Math.
198
(2014), 701–737], 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.