Number of integers represented by families of binary forms (I)

Abstract

We consider some families of binary binomial forms $aX^d+bY^d$, with $a$ and $b$ integers. Under suitable assumptions, we prove that every rational integer $m$ with $|m|\ge 2$ is only represented by a finite number of the forms of this family (with varying $d,a,b$). Furthermore {the number of such forms of degree $\ge d_0$ representing $m$ is bounded by $O(|m|^{(1/d_0)+\epsilon})$} uniformly for $\vert m \vert \geq 2$. We also prove that the integers in the interval $[-N,N]$ represented by one of the form of the family with degree $d\geq d_0$ are almost all represented by some form of the family with degree $d=d_0$. In a previous {paper} we investigated the particular case where the binary binomial forms are positive definite. We now treat the general case by using a lower bound for linear forms of logarithms.