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\(p\)-adic logarithmic forms and group varieties. II. (English) Zbl 0928.11031
Let \(K\) be a number field of degree \(d\), \(\alpha_{1},\ldots,\alpha_{n}\) non zero elements in \(K\), \(b_{1},\ldots,b_{n}\) rational integers such that the number \(\Xi=\alpha_{1}^{b_{1}}\cdots\alpha_{n}^{b_{n}} -1\) is not zero. Let \(\mathfrak p\) be a prime ideal in the ring of integers of \(K\) above a prime number \(p\). The goal is to establish an explicit upper bound for \(\text{ord}_{\mathfrak p}(\Xi)\), namely the exponent to which \(\mathfrak p\) divides the principal fractional ideal \((\Xi)\) in \(K\). Denote by \(h\) the absolute logarithmic height and define \(h_{1},\ldots,h_{n}\) by \[ h_{j}=\max\{h(\alpha_{j}),\log p\} \qquad (1\leq j\leq n). \] Let \(B\geq 3\) be a positive integer such that \(| b_{i}| \leq B\) for \(1\leq i\leq n\). As a special case of his main estimate, the author deduces \[ \text{ord}_{\mathfrak p}(\Xi)< C(n,d,{\mathfrak p})h_{1}\cdots h_{n}\log B, \] where \[ C(n,d,{\mathfrak p})=19\bigl(20\sqrt{n+1}d\bigr)^{2(n+1)} e_{\mathfrak p}^{n-1}\cdot {{\mathfrak p}^{{f_{\mathfrak p}}}\over (f_{\mathfrak p}\log {\mathfrak p})^{2}} \cdot \log(e^{5}nd), \] \(e_{\mathfrak p}\) is the ramification index and \(f_{\mathfrak p}\) the residue class degree of \({\mathfrak p}\).
This is a sharpening of the author’s previous result [J. Reine Angew. Math. 502, 29-92 (1998; Zbl 0912.11029)]. The refinement is due to the fact that the author succeeds to adapt to the \(p\)-adic case an argument of E. M. Matveev [Izv. Ross. Akad. Nauk, Ser. Mat. 62, No. 4, 81-136 (1998), Engl. translation in Izv. Math. 62, No. 4, 723-772 (1998; Zbl 0923.11107)].
Such estimates have a wide range of applications; therefore one should stress that for practical purposes (for instance for solving diophantine equations), as a general rule, it is much more efficient to apply the more elaborate statements in the paper under review than the special above mentioned easy case. For instance a much smaller value for \(C(n,d,\mathfrak p)\) is achieved when \[ [K(\alpha_{0}^{1/q},\alpha_{1}^{1/q},\ldots, \alpha_{n}^{1/q}):K]=q^{n+1}, \] where \[ q=\begin{cases} 2 & \text{if } p>2 ,\\ 3 & \text{if } p=2, \end{cases} \] and \(\alpha_{0}\) is a generator of the torsion subgroup of \(K^{\times}\).
In a forthcoming joint paper of the author with C. L. Stewart, they improve their result related to the \(abc\) conjecture, replacing an exponent \(2/3\) by \(1/3\) [Math. Ann. 291, No. 2, 225-230 (1991; Zbl 0761.11030)].

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