A simultaneous approximation problem for exponentials and logarithms
Veekesh Kumar, Riccardo Tosi
TL;DR
This work extends Brownawell-type results to the simultaneous approximation problem for the triple $( rac{ ext{log } α_2}{ ext{log } α_1}, α_1^β, α_2^β)$ with β a quadratic irrational and α1, α2 multiplicatively independent algebraic numbers. By constructing a carefully controlled auxiliary polynomial and employing height/degree estimates, Schwarz-type bounds, and semi-resultants, the authors obtain lower bounds and a quantitative measure of algebraic independence among the three numbers. The approach yields explicit bounds showing that values of two coprime polynomials at the specified point cannot be too small, and it culminates in a robust algebraic-independence result with effective constants. These results advance the understanding of simultaneous approximation phenomena involving logarithmic and exponential expressions in the setting of algebraic numbers, with implications for transcendence theory and Diophantine approximation.
Abstract
Let $α_1,α_2$ be non-zero algebraic numbers such that $\frac{\log α_2}{\logα_1}\notin\mathbb{Q}$ and let $β$ be a quadratic irrational number. In this article, we prove that the values of two relatively prime polynomials $P(x,y,z)$ and $Q(x,y,z)$ with integer coefficients are not too small at the point $\left(\frac{\logα_2}{\log α_1},α_1^β, α_2^β\right)$. We also establish a measure of algebraic independence of those numbers among $\frac{\logα_2}{\log α_1}$, $α^β_1$ and $α^β_2$ which are algebraically independent.