Diophantine approximations in the field of p-adic numbers (Q800404)

A theorem of the following type is proved. Suppose \(\vartheta_ 0,\vartheta_ 1,\vartheta_ 2\) are elements of the p-adic field \({\mathbb{Q}}_ p\), and there exist numbers \(\kappa\),c,C with \(\kappa >0\), \(C>c>0\) such that for all sufficiently large N we can find a homogeneous polynomial \(P_ N(x_ 0,x_ 1,x_ 2)\) in \({\mathbb{Z}}[x_ 0,x_ 1,x_ 2]\), of ''taille'' at most N, with the property that \(\exp (- CN^{\kappa})\leq| P_ N(\vartheta_ 0,\vartheta_ 1,\vartheta_ 2)|_ p\leq\exp (-cN^{\kappa}).\) Then \(\kappa\leq 3.\) This can be viewed as a generalization to two variables of Gel'fond's well-known transcendence criterion, and the upper bound for the exponent is presumably best possible. Earlier \textit{E. Reyssat} [J. Reine Angew. Math. 329, 66-81 (1981; Zbl 0459.10023)], following work of G. V. Chudnovsky, had proved a version in n variables valid even in the Archimedean case, but with exponent bound \(2^ n.\) The author proves his result by using the elimination techniques of his fundamental paper [Izv. Akad. Nauk SSSR, Ser. Mat. 41, 253-284 (1977; Zbl 0354.10026); translated as Math. USSR, Izv. 11, 239-270 (1977)] to construct useful definitions of the ''taille'' of an ideal and the value of an ideal at a point. [It should be noted that recently \textit{P. Philippon}, in a paper to appear in Publ. Math., Inst. Hautes Étud. Sci., has developed the author's ideas further, following work of \textit{M. Waldschmidt} and \textit{Y. Zhu} [C. R. Acad. Sci., Paris, Sér. I 297, 229-232 (1983; Zbl 0531.10037)] and \textit{Y. Zhu} [C.R. Math. Acad. Sci., Soc. R. Can. 6, 297- 302 (1984)]. Among other things, he proves a criterion in n variables which implies the sharp exponent bound \(n+1\) in general.]