Prime solutions to polynomial equations in many variables and differing degrees
Abstract
Let f = (f1, …, fR) 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 fj (x1, …, xn) = 0 \ (1 ≤ j ≤ 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 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.