Lower bounds for linear forms in two p-adic logarithms

We prove explicit lower bounds for linear forms in two p-adic logarithms. More specifically, we establish explicit lower bounds for the p-adic distance between two integral powers of algebraic numbers, that is, $|\Lambda {|}_p=|{\alpha }_1^{b_1}-{\alpha }_2^{b_2}{|}_p$ (and corresponding explicit upper bounds for $v_p\left(\Lambda \right)$), where ${\alpha }_1,{\alpha }_2$ are numbers that are algebraic over $Q$ and $b_1,b_2$ are positive rational integers.

This work is a p-adic analogue of Gouillon's explicit lower bounds in the complex case. Our upper bound for $v_p\left(\Lambda \right)$ has an explicit constant of reasonable size and the dependence of the bound on B (a quantity depending on $b_1$ and $b_2$) is $\log B$, instead of ${(\log B)}^2$ as in the work of Bugeaud and Laurent in 1996.