The number of solutions to the trinomial Thue equation

Abstract

In this paper, we study the number of integer pair solutions to the equation $|F(x,y)| = 1$ where $F(x,y) \in \Z[x,y]$ is an irreducible (over $\Z$) binary form with degree $n \geq 3$ and exactly three nonzero summands. In particular, we improve Thomas' explicit upper bounds on the number of solutions to this equation (see [13]). For instance, when $n \geq 219$, we show that there are no more than 32 integer pair solutions to this equation when $n$ is odd and no more than 40 integer pair solutions to this equation when $n$ is even, an improvement on Thomas' work in [13], where he shows that there are no more than 38 such solutions when $n$ is odd and no more than 48 such solutions when $n$ is even.