$S$-parts of values of univariate polynomials,\\ binary forms and decomposable forms at integral points

Abstract

To Robert Tijdeman on his 75-th birthday 1 n + ǫ for every ǫ > 0 and instead of κ 2 an ineffective number depending on f (X), S and ǫ. This is in fact an easy application of the p-adic Thue-Siegel-Roth Theorem. We show that the exponent 1 n is best possible.

‐parts of Terms of Integer Linear Recurrence Sequences

“…We keep the above notation throughout the present paper. The case t = 1, that is, of sequences (f (n)a n ) n≥0 where f (X) is an integer polynomial and a a non-zero integer, can be treated using the work of [4,1]. Thus, in all what follows, we assume that t ≥ 2, the polynomials f 1 (X), .…”

Effective simultaneous rational approximation to pairs of real quadratic numbers

“…It is ultimately a consequence of the quantity B ′ , which has its origin in Feldman's papers [13,14] and is the key tool for his effective improvement of Liouville's bound; see Theorem 2.1 and the discussion below it. Other consequences of the quantity B ′ can be found in [10] and in the recent papers [9,11,12].…”

S-parts of values of univariate polynomials

“…Motivated by previous work of Gross and Vincent ([GV13]), Bugeaud, Evertse and Győry proved in [BEG18] that if f ∈ Z[X] is a polynomial of degree n ≥ 1 without multiple roots, then for any δ > 0 and any x ∈ Z with f (x) = 0 one has…”

“…Going through the proof of theorem A in [BEG18], it is not difficult to realize that the polynomial factor and the logarithmic factor in the asymptotic rate of N (f, S, ε, B) as B → ∞ have a very different nature. If S ′ = {p}, then the rate of N (f, S, ε, B) as B → ∞ is polynomial with exponent independent of the specific prime p, fact that is intimately related to the existence of an elementary asymptotic rate for N (f, S, ε, B) as B → ∞ in the case #S ′ ≥ 2.…”

“…The techniques in this paper can be adapted to the similar problems considered in [BEG18] in the context of decomposable forms. This leads to significant improvements on the corresponding results in [BEG18]. We will present our results on decomposable forms in a subsequent paper.…”