REPRESENTATION OF SMALL INTEGERS BY BINARY FORMS

Abstract

We establish some upper bounds for the number of integer solutions to the Thue inequality |$|F(x , y)| \leq m$|⁠, where |$F$| is a binary form of degree |$n \geq 3$| 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 |$m < |D|^{{1}/{2(n-1)}}$|⁠.