On the k-free values of the polynomial xy^k+ C

Consider the polynomial f(x, y) = xy^k+ C for k≥ 2 and any nonzero integer constant C. We derive an asymptotic formula for the k-free values of f(x, y) when x, y≤ H. We also prove a similar result for the k-free values of f(p, q) when p, q≤ H are primes, thus extending Erdős’ conjecture for our specific polynomial. The strongest tool we use is a recent generalization of the determinant method due to Reuss.