Measure of algebraic independence for almost all pairs of p-adic numbers (Q1065063)

It is proved that ''almost all p-adic fields of transcendence degree 2 over \({\mathbb{Q}}\) have strict transcendence type 3''. More precisely, for almost all \((\vartheta_ 1,\vartheta_ 2)\) in \({\mathbb{Q}}^ 2_ p\) there exists a positive constant \(c=c(\vartheta_ 1,\vartheta_ 2)\) such that for each \(t\geq 1\) and every non-zero polynomial R in \({\mathbb{Z}}[x_ 1,x_ 2]\), of degree at most t with coefficients of logarithmic absolute values at most t, we have \(| R(\vartheta_ 1,\vartheta_ 2)|_ p\geq \exp (-ct^ 3).\) The exponent 3 here is well-known to be best possible. The corresponding results in one variable, for \({\mathbb{R}}\) as well as \({\mathbb{Q}}_ p\), were proved by the author some time ago [Mat. Zametki 15, 405-414 (1974; Zbl 0287.10022)]. The present proofs are based on the author's fundamental paper [Izv. Akad. Nauk SSSR, Ser. Mat. 41, 253-284 (1977; Zbl 0354.10026)] together with his more recent article [Mat. Zametki 35, 653-662 (1984; Zbl 0549.10024)].