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\(p\)-adic logarithmic forms and group varieties. I. (English) Zbl 0912.11029
Let \(\alpha_{1},\ldots,\alpha_{n}\) be non zero algebraic numbers in a number field \(K\) of degree \(d\). In the ring \({\mathcal O}_{K}\) of integers of \(K\), let \({\mathfrak p}\) be a prime ideal above a prime number \(p\). Denote by \(f_{\mathfrak p}\) the residue class degree of \({\mathfrak p}\). For \(\alpha\in K\), \(\alpha\not=0\), denote by \(\text{ord}_{\mathfrak p}(\alpha)\) the exponent to which \({\mathfrak p}\) divides the principal fractional ideal \((\alpha)\). Further denote by \(h(\alpha)\) the absolute logarithmic height of \(\alpha\), namely \[ h(\alpha)=\delta{\mathfrak p}{-1}\left(\log a_0+ \sum_{i=1}{\mathfrak p}{\delta} \log\max\{1,| \alpha^{(i)}| \}\right), \] where the minimal polynomial of \(\alpha\) over \(\mathbb Z\) is \(a_{0}X{\mathfrak p}{\delta}+\cdots+a_{\delta}= a_{0}\prod_{i=1}^{\delta} (X-\alpha{\mathfrak p}{(i)})\). Define \(h_{1},\ldots,h_{n}\) by \(h_{j}=\max\{h(\alpha_{j}),\log p\}\) \((1\leq j\leq n)\). Let \(b_{1},\ldots,b_{n}\) be rational integers such that the number \(\Xi=\alpha_{1}{\mathfrak p}{b_{1}}\cdots\alpha_{n}^{b_{n}} -1\) does not vanish. Let \(B\geq 3\) be a positive integer such that \(| b_{i}| \leq B\) for \(1\leq i\leq n\). A simple special case of the author’s main result is \[ \text{ord}_{\mathfrak p}(\Xi)<C(n,d,{\mathfrak p})h_{1}\cdots h_{n}\log B, \] where \[ C(n,d,{\mathfrak p})=12\left({6(n+1)d\over \sqrt{\log p}}\right) {\mathfrak p}{2(n+1)}\bigl(p{\mathfrak p}{f_{\mathfrak p}}-1\bigr) \log(e^{5}nd). \] This is a \(p\)-adic analog of a result of A. Baker and G. Wüstholz [J. Reine Angew. Math. 442, 19-62 (1993; Zbl 0788.11026)]. Among the previous measures of linear independence for \(p\)-adic logarithms of algebraic numbers, the only one which is not included in the author’s main result is the estimate of Y. Bugeaud and M. Laurent, which deals with only two logarithms [J. Number Theory 61, No. 2, 311-342 (1996; Zbl 0870.11045)].
In a forthcoming paper [K. Yu, \(p\)-adic logarithmic forms and group varieties. II, Acta Arith. (to appear)], the author improves his estimate (at least in some cases, among the most important ones for applications) by including an idea of E. M. Matveev’s [Explicit lower estimates for rational homogeneous linear forms in logarithms of algebraic numbers, Izv. Akad. Nauk SSSR, Ser. Mat. 62, No. 4, 81-136 (1998) (English translation in Izv. Math. 62, No. 4, 723-772 (1998)].

11J86 Linear forms in logarithms; Baker’s method
11J61 Approximation in non-Archimedean valuations
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