A simultaneous approximation problem for exponentials and logarithms

TL;DR

This paper establishes lower bounds and algebraic independence measures for certain exponential and logarithmic values involving algebraic numbers and quadratic irrationals, advancing understanding of their approximation properties.

Contribution

It proves non-trivial lower bounds for polynomial values at specific exponential-logarithmic points and introduces algebraic independence measures for these numbers.

Findings

Lower bounds for polynomial values at given points.

Measure of algebraic independence among key numbers.

Results applicable to algebraic and transcendental number theory.

Abstract

Let $\alpha_1,\alpha_2$ be non-zero algebraic numbers such that $\frac{\log \alpha_2}{\log\alpha_1}\notin\mathbb{Q}$ and let $\beta$ 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 $\left(\frac{\log\alpha_2}{\log \alpha_1},\alpha_1^\beta, \alpha_2^\beta \right)$. We also establish a measure of algebraic independence of those numbers among $\frac{\log\alpha_2}{\log \alpha_1}$, $\alpha^\beta_1$ and $\alpha^\beta_2$ which are algebraically independent.