Effective approximation to complex algebraic numbers by quadratic numbers
Prajeet Bajpai, Yann Bugeaud
TL;DR
The paper addresses the problem of effectively approximating complex non-real algebraic numbers $\xi$ by quadratic complex algebraic numbers $\alpha$, improving Liouville-type bounds for degrees $d\ge 4$ under not-a-CM-field assumptions when $d=4$. It employs Baker's theory of linear forms in logarithms to obtain explicit, effective lower bounds of the form $|\xi-\alpha| > c(\xi) H(\alpha)^{-d/2+\kappa(\xi)}$, by linking the approximation problem to lower bounds for $|\mathrm{Norm}_{K/\mathbb{Q}}(P_{\alpha}(\xi))|$ and analyzing the unit structure in $K=\mathbb{Q}(\xi)$ via a normalization $\mu=x/u$. The method distinguishes two embedding scenarios (one where $\sigma_d(x)=x$ and one where it does not) and uses linear forms in logarithms to control heights and exponents, yielding an effective exponent $\kappa(\xi)$ (and a corresponding constant $c(\xi)$). This extends previous results for totally complex $\xi$ and highlights the role of norm-form equations and unit theory in deriving effective diophantine bounds for non-totally complex fields. The outcome sharpens the Liouville-type inequality with explicit constants under precise structural assumptions on $\xi$ and its field, contributing to concrete effective approximation results in algebraic number theory.
Abstract
We establish an effective improvement on the Liouville inequality for approximation to complex non-real algebraic numbers by quadratic complex algebraic numbers.