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Linear forms in logarithms in the p-adic case. (English) Zbl 0656.10028
New advances in transcendence theory, Proc. Symp., Durham/UK 1986, 411-434 (1988).
[For the entire collection see Zbl 0644.00005.]
The article is dedicated to the correction of several mistakes which the author found in A. J. van der Poorten’s paper (of the same title) [Transcend. Theory, Proc. Conf., Cambridge 1976, 29-57 (1977; Zbl 0367.10034)]. Let K be a number field, let P be a prime ideal in K lying above a rational p. Let \(\alpha_ 1,...,\alpha_ n\in K\), and let \(b_ 1,...,b_ n\) be rational integers \(\leq B\). Van der Poorten’s theorems claimed that \(Ord_ P(\alpha_ 1^{b_ 1}...\alpha_ n^{b_ n}- 1)\) is either infinite or bounded above by a function that involves the heights of \(\alpha_ 1,...,\alpha_ n\) and Log B.
The article goes over the proofs given by A. J. van der Poorten and it proposes sometimes another demonstration, sometimes a lemma slightly different but sufficient for the continuation. For example, in an extrapolation procedure he gives an analogue of van der Poorten’s lemma 4 and he sets up a simple counter-example showing the lemma be false.
The study seems to be done very carefully. However, the author announced the full details to appear in another article [see Max Planck Institut für Mathematik, Preprint 87-20].
Reviewer: A.Escassut

11J81 Transcendence (general theory)