Counting squarefree values of polynomials with error term

It was shown by Granville that the ABC conjecture allows one to prove asymptotic estimates on the number of squarefree values of polynomials. However, his proof gives no information on the error term of the asymptotic formula. On the ABC conjecture, we prove an asymptotic formula with error term using a different technique. From the ABC conjecture we also deduce an asymptotic formula with error term for the number of squarefree values of polynomials on certain sets of integers that are residually well distributed in a suitable sense.