Linear Equations in Variables which Lie in a Multiplicative Group

Abstract

Let K be a field of characteristic 0 and let n be a natural number. Let Γ be a subgroup of the multiplicative group (K *

The Dynamical Mordell–Lang Conjecture

“…Moreover, the Skolem-Mahler-Lech theorem is used in the proof of [GTa]. Evertse, Schlickewei, and Schmidt [ESS02] have given a strong quantitative version of the Skolem-Mahler-Lech theorem. It may be possible to use their result to give more precise versions of the theorems of this paper.…”

Conditional results on the birational section conjecture over small number fields

“…Suppose that condition (2) is satisfied. Then, to verify the implication (2) ⇒ (1), by the verified implication (3) ⇒ (2) [in the case where we take "k " to be k], it suffices to verify that, for any open subgroup…”

Diophantine equations E(x) = P(x) with E exponential, P polynomial

“…For instance, take λ = {1, 2, 3}. According to [1] (see also the formulations in Section 2 of [5]), the solutions in S(λ) fall into 1 classes, and for solutions in a given class the triples (α 2x 1 , −α · α 3x 2 , β x 3 ) are proportional to a given triple, i.e., will have α 2x 1 = γ(−α · α 3x 2 ) = γ β x 3 for some γ, γ . But these relations for fixed γ, γ have (by the multiplicative independence of α, β) at most one solution in integers x 1 , x 2 , x 3 .…”

“…We will show that for any partition P not containing Λ 0 , |S(P)| 1. When P is no proper partition, so that for (x, y) ∈ S(P) no proper subsum of (3.1) vanishes, then by [1], the solutions in S(P) fall into 1 classes, with solutions in a given class having β x = γ 1 β y = γ 2 β (σ)x = γ 3 β (σ)y with fixed γ 1 , γ 2 , γ 3 . By the multiplicative independence of β, β (σ) , there can be at most one such pair (x, y).…”

“…We thus may suppose that for some set Λ ∈ P, two summands g 1li α x l li and g 1mj α x m mj with (l, i) = (m, j) and nonzero g 1li , g 1mj belong to Λ. Invoking [1] we see that solutions in S(Λ) fall into 1 classes, and g 1mj α x m mj = γg 1li α x l li with fixed γ for solutions x in a given class.…”