Linear forms in two logarithms and interpolation determinants. (English) Zbl 0801.11034

Laurent, Michel

Acta Arith. 66, No. 2, 181-199 (1994).

The author provides a precise lower bound for the absolute value of a linear combination of two logarithms of real algebraic numbers with integer coefficients. This lower bound is explicit and improves in the real case an earlier result of M. Mignotte and M. Waldschmidt [Ann. Fac. Sci. Toulouse Math. 97, 43-75 (1989; Zbl 0702.11044)] essentially by reducing the multiplicative constant from 270 to 87 or to an even smaller number (see the precise results below). For the applications, it also compares with the general lower bound established by A. Baker and G. Wüstholz [J. Reine Angew. Math. 442, 19- 62 (1993; Zbl 0788.11026)] where the dependency on the absolute values of the coefficients is better at the expense of a constant \(>10^ 8\).
The method of proof also is interesting. It involves the technique of interpolation determinants developed by M. Laurent [Astérisque 198-200, 209-230 (1991; Zbl 0762.11027)]. This technique avoids the use of Siegel’s lemma by constructing a determinant for which precise lower and upper bounds are given. The result follows as usual by comparing those bounds.
Let \(\alpha_ 1\), \(\alpha_ 2\) be real algebraic numbers which are assumed to be \(\geq 1\) and multiplicatively independent. Choose real numbers \(a_ 1, a_ 2>1\) such that \(h(\alpha_ i)\leq \log a_ i\) for \(i=1,2\), where \(h(\alpha_ i)\) stands for the usual Weil absolute logarithmic height of \(\alpha_ i\). Also, let \(b_ 1\), \(b_ 2\) be positive rational integers. Put \[ \Lambda= b_ 2\log \alpha_ 2- b_ 1\log \alpha_ 1 \qquad \text{and} \qquad b'= {{b_ 1} \over {D\log a_ 2}} + {{b_ 2} \over {D\log a_ 1}}. \] The author proves a general result from which he deduces:
For each constant \(c>48\), there exists a number \(b'(c)\) such that, if \(b'\geq b'(c)\), then \[ \log |\Lambda |\geq -cD^ 4 (\log b')^ 2 \log a_ 1 \log a_ 2. \] In particular, if \(\log a_ 1\geq 1\), \(\log a_ 2\geq 1\) and \(\log b'\geq 25,\) then \[ \log | \Lambda|\geq - 87D^ 4 (0.5+ \log b')^ 2 \log a_ 1 \log a_ 2. \] A preliminary version of this paper can also be found in the appendix of M. Waldschmidt [Linear independence of logarithms of algebraic numbers (French) (IMSc. Report No. 116, The Institute of Mathematical Sciences, Madras) (1992)].

Reviewer: D.Roy (Ottawa)

MSC:

11J86

Linear forms in logarithms; Baker’s method

Keywords:

linear forms in logarithms; height; lower bound; linear combination of two logarithms of real algebraic numbers with integer coefficients; interpolation determinants

Citations:

Zbl 0702.11044; Zbl 0788.11026; Zbl 0762.11027