On a conjecture of Levesque and Waldschmidt

Abstract

One of the first parametrised Thue equations, $$\left| X^3 - (n-1)X^2 Y - (n+2) XY^2 - Y^3 \right| = 1,$$ over the integers was solved by E. Thomas in 1990. If we interpret this as a norm-form equation, we can write this as $$\left| N_{K/\mathbb{Q}}\left( X - \lambda_0 Y \right) \right| = \left| \left( X-\lambda_0 Y \right) \left( X-\lambda_1 Y \right) \left( X-\lambda_2 Y \right) \right| =1$$ if $\lambda_0, \lambda_1, \lambda_2$ are the roots of the defining irreducible polynomial, and $K$ the corresponding number field.\par\medskip Levesque and Waldschmidt twisted this norm-form equation by an exponential parameter $s$ and looked, among other things, at the equation $$\left| N_{K/\mathbb{Q}}\left( X - \lambda_0^s Y \right) \right| = 1.$$ They solved this effectively and conjectured that introducing a second exponential parameter $t$ and looking at $$\left| N_{K/\mathbb{Q}}\left( X - \lambda_0^s\lambda_1^t Y \right) \right| = 1$$ does not change the effective solvability. \par\medskip We want to partially confirm this, given that $$\min\left( \left| 2s-t \right|, \left| 2t-s \right|, \left| s+t \right| \right) > \varepsilon \cdot \max\left( \left|s\right|, \left|t\right| \right) > 2,$$ i.e. the two exponents do not almost cancel in specific cases.