Multiplicative dependence in the sumset of multiplicative groups

2026/07/30 by Yuri Bilu, Florian Luca
Mathematics · #math.NT

paper · pdf

Preliminary version, comments are welcome

arxiv created 2026/07/30 · arxiv updated 2026/08/03

Abstract

Let Γ and Δ be finitely generated multiplicative groups of algebraic numbers such that Γ∩Δ is a finite group. We show that, up to finitely many exceptions, non-zero sums x₁+y₁ and x₂+y₂, with x₁, x₂∈ Γ and y₁,y₂∈ Δ, are multiplicatively dependent only if x₁/x₂=y₁/y₂ is a root of unity. For m≥ 3, we discuss possible shapes of m multiplicatively dependent sums x₁+y₁, …, x_m+y_m with x₁, …, x_m ∈ Γ and y₁, …, y_m ∈ Δ. For m=3 we classify such sums, up to finitely many exceptions, assuming the abc-conjecture.

Citations

• Explicit bounds for the solutions of superelliptic equations over number fields
• Binary polynomial power sums vanishing at roots of unity
• A lower bound for the height of a rational function at S-unit points