Number of integers represented by families of binary forms (II): binomial forms
Volume 214 / 2024
Abstract
We consider some families of binary binomial forms , 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 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 forms of the family of degree d\geq d_0 are almost all represented by some form of the family of degree d=d_0 if such forms of degree d_0 exist.
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 in logarithms.
