On algebraic relations in non-archimedean fields (Q1316815)

The author proves the following effective theorem: suppose that the series \(f_ 1(z), \dots, f_ m(z)\) belong to the class \(F(\mathbb K, c_ 1, c_ 2, c_ 3, q_ 0)\) and form a solution of a Siegel normal system. Let \(\xi \in \mathbb K\), \(\xi T(\xi) \neq 0\), \(\delta>0\). Then there exists an effective constant \(H_ 0\) such that for any polynomial \(Q(y_ 1, \dots, y_ n) \not \equiv 0\) of degree \(d\) and the height \(\leq H\), there exists for \(H \geq H_ 0\) a prime number \(p\) such that \[ p \leq (H \delta) {m + \delta \choose m} {m \log H \over \log \log H} \] and a place \(v \mid p\) such that in \(\mathbb K_ v\) the relation \(Q(f_ 1(\xi), \dots, f_ m (\xi)) \neq 0\) holds. The author also gives an application to hypergeometric series. The proof uses a modification of Siegel-Shidlovskii and the work by \textit{F. Beukers}, \textit{W. D. Brownawell} and \textit{G. Heckman} [Ann. Math. (2) 127, 279--308 (1988; Zbl 0652.10027)].