\(p\)-adic continued fractions and algebraic independence (Q1301688)

Let \(k\in \mathbb{N}\), \(k\geq 2\). for \(2\leq j\leq k\), let \(A_j= [a_{0,j}, a_{1,j},\dots, a_{n,j},\dots]\) be \(p\)-adic continued fractions. The author proves the following theorem: Suppose there exist \(r>1\) and \(\alpha>2\) such that \(r^{-1}|a_{n,j}|_p> |a_{n,j-1}|_p\) for \(j= 2,\dots, k\) and \(|a_{n,1}|_p> |a_{n-1,k}|_p^2\) for every large \(n\); then for every \(P\in \mathbb{Z} [X_1,\dots, X_n]\) of total degree less than \(\frac{\alpha}{2}\), the \(p\)-adic number \(P(A_1,\dots, A_n)\) is transcendental. The proof rests on \textit{D. Ridout}'s theorem [Mathematika, London 5, 40-48 (1958; Zbl 0085.03501)].