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Corrections to “How to explicitly solve a Thue-Mahler equation”. (English) Zbl 0791.11011

From the text: In Appendix A3 of our paper [ibid. 84, No. 3, 223-288 (1992; Zbl 0773.11023)] we make use of Corollary 2 of [BGMMS] which contains a serious misprint [see the correction in this volume ( Zbl 0791.11033)]. As a consequence, our constant \(c_ 7\) defined on p. 284, should be multiplied by \(m^{2n + 1}\). It will be clear that a larger upper bound for \(B\) can be derived from the corrected result of [BGMMS], and that the general method of our paper is insensitive to the actual value of the constants. However, in any particular example the computations do of course depend on the correct value of the constants. Therefore we have to reconsider the details of our example, treated in the \(^{\text{Ex}}\)-sections (stated in these corrections).
Recently A. Baker and G. Wüstholz [J. Reine Angew. Math. 442, 19-62 (1993; 788.11026)] proved a new lower bound for linear forms in logarithms of algebraic numbers, which is considerably sharper than the one given in our Appendix A3 and found that they can take \(c_ 7 = 2.2044 \times 10^{38}\), \(c_ 8 = 0\). This leads, by taking \(c_{16} = 10^{-9}\), again to \(c_{\text{real}} < 9.844 \times 10^{49}\), which is exactly the upper bound found in the paper. This shows that the reduction procedure as worked out in the Section \(15^{\text{Ex}}\) and \(16^{\text{Ex}}\) is in fact adequate to prove the main result on our particular Thue-Mahler equation. Finally three minor misprints are corrected.

MSC:

11D41 Higher degree equations; Fermat’s equation
11J85 Algebraic independence; Gel’fond’s method
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References:

[1] J. Blass , A.M.W. Glass , D.K. Manski , D.B. Meronk and R.P. Steiner : Constants for lower bounds for linear forms in logarithms of algebraic numbers II. The homogeneous rational case , Acta Arithmetica 55 (1990) 15-22. · Zbl 0709.11037
[2] A. Baker and G. Wüstholz : Logarithmic forms and group varieties , Journal für die reine und angewandte Mathematik, (1993) to appear. · Zbl 0788.11026 · doi:10.1515/crll.1993.442.19
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