Linear forms in p-adic logarithms.

*(English)*Zbl 0699.10050Let v be the usual valuation in \({\mathbb{C}}_ p\). Let \(\alpha_ 1,...,\alpha_ n\) be non zero p-adic numbers in an algebraic number field K of degree D over \({\mathbb{Q}}\), such that the height of \(\alpha_ j\) does not exceed \(A_ j\geq e^ e\) (1\(\leq j\leq n)\). Many results were obtained on the linear forms in p-adic logarithm log \(\alpha_ j\) by Mahler, Gel’fond, Schinzel, Baker, Coates, Sprindzhuk, Kaufman.

In 1977, A. J. van der Poorten obtained two theorems that said roughly: “there is no n-tuple \((b_ 1,...,b_ n)\in {\mathbb{Q}}^ n\) such that \(| b_ j| \leq B\) and such that \(v(\alpha_ 1^{b_ 1},...,\alpha_ n^{b_ n}-1)\) is greater than a big expression involving the \(A_ i\) and B.” The author explains that there are several errors and inaccuracies shortly described at the end of the article, in the proofs of these theorems. Now he gives two very technical theorems that imply van der Poorten’s theorems.

The way consists of directly giving \(\alpha_ 1^{b_ 1},...,\alpha_ n^{b_ n}-1\) \((b_ j\in {\mathbb{Q}})\) an upper bound (the first theorem also supposes that \([K(\alpha_ 1^{1/q},...,\alpha_ n^{1/q}) : K]=q^ n\) for some \(q\in {\mathbb{N}}^*).\)

In p-adic analysis, the author mainly uses very classical and elementary properties of the analytic elements in a closed disk, and then obtains an interesting Lemma for extrapolation: Let F be an analytic element in the unit disk d(0,1), satisfying \(\| F\|_{d(0,1)}\leq 1,\) and additional conditions on the derivatives at the point \(sp^{\theta}\) with \(\theta\in {\mathbb{Q}}\), \(s\in {\mathbb{N}}^*\), \(s\leq R\) and q a prime number \((q\neq p)\); then v \((F(\ell p^{\theta}/q))\) has a lower bound in the form \((1-1/q)\) RMO whenever \(\ell \in {\mathbb{Z}}.\)

All the demonstrations are very long and much computational and represent a very big work.

In 1977, A. J. van der Poorten obtained two theorems that said roughly: “there is no n-tuple \((b_ 1,...,b_ n)\in {\mathbb{Q}}^ n\) such that \(| b_ j| \leq B\) and such that \(v(\alpha_ 1^{b_ 1},...,\alpha_ n^{b_ n}-1)\) is greater than a big expression involving the \(A_ i\) and B.” The author explains that there are several errors and inaccuracies shortly described at the end of the article, in the proofs of these theorems. Now he gives two very technical theorems that imply van der Poorten’s theorems.

The way consists of directly giving \(\alpha_ 1^{b_ 1},...,\alpha_ n^{b_ n}-1\) \((b_ j\in {\mathbb{Q}})\) an upper bound (the first theorem also supposes that \([K(\alpha_ 1^{1/q},...,\alpha_ n^{1/q}) : K]=q^ n\) for some \(q\in {\mathbb{N}}^*).\)

In p-adic analysis, the author mainly uses very classical and elementary properties of the analytic elements in a closed disk, and then obtains an interesting Lemma for extrapolation: Let F be an analytic element in the unit disk d(0,1), satisfying \(\| F\|_{d(0,1)}\leq 1,\) and additional conditions on the derivatives at the point \(sp^{\theta}\) with \(\theta\in {\mathbb{Q}}\), \(s\in {\mathbb{N}}^*\), \(s\leq R\) and q a prime number \((q\neq p)\); then v \((F(\ell p^{\theta}/q))\) has a lower bound in the form \((1-1/q)\) RMO whenever \(\ell \in {\mathbb{Z}}.\)

All the demonstrations are very long and much computational and represent a very big work.

Reviewer: A.Escassut