Lower bounds for linear forms in two \(p\)-adic logarithms (Q6623039)

Given two algebraic numbers \(\alpha_1,\alpha_2\), let \(\Lambda=\{\alpha_1^{b_1}-\alpha_2^{b_2}:b_1,b_2\in\mathbb{Q}^+\}\). A well-studied problem is computing \(\vert \Lambda\vert=\min_{\alpha\in \Lambda}\vert\alpha\vert\). In this paper, the author considers the \(p\)-adic absolute value \(\vert \Lambda\vert_p=\min\{\vert \alpha\vert_p:\alpha\in \Lambda\}\). In particular, the author bounds from above the \(p\)-adic valuation of \(\Lambda\) with respect to the \(\log(h(\alpha_1))\log(h(\alpha_2))\) and \([\mathbb{Q}(\alpha_1,\alpha_2):\mathbb{Q}]\).\N\NTo prove this bound, the author shows that a certain algebraic condition implies this bound. The author then proves that the algebraic condition holds by bounding the \(p\)-adic norm of an interpolation determinant from above and below. The upper bound is obtained by using a \(p\)-adic version of Schwartz's lemma, whereas the lower bound is obtained through a \(p\)-adic version of Liouville's estimate.