Number of integers represented by families of binary forms II: binomial forms

2023/06/04 by Fouvry, Étienne, Waldschmidt, Michel · 1 citation
#11E76 11D45 11D85 #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.2306.02462

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|≥ 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 ≥ d₀ representing m is bounded by O(|m|^(1/d₀)+ε) uniformly for \vert m \vert ≥ 2. We also prove that the integers in the interval [-N,N] represented by one of the form of the family with degree d≥ d₀ are almost all represented by some form of the family with degree d=d₀. 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.

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