Representation of Small Integers by Binary Forms

We establish some upper bounds for the number of integer solutions to the Thue inequality |F(x, y)| ≤ m, where F is a binary form of degree n ≥ 3 and with non-zero discriminant D, and m is an integer. Our upper bounds are independent of m, when m is smaller than |D|^1/4(n-1). We also consider the Thue equation |F(x, y)| = m and give some upper bounds for the number of its integral solutions. In the case of equation, our upper bounds will be independent of integer m, when m |D|^1/2(n-1).