# Linear forms in two logarithms and interpolation determinants

Acta Arithmetica (1994)

- Volume: 66, Issue: 2, page 181-199
- ISSN: 0065-1036

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topMichel Laurent. "Linear forms in two logarithms and interpolation determinants." Acta Arithmetica 66.2 (1994): 181-199. <http://eudml.org/doc/206599>.

@article{MichelLaurent1994,

abstract = {
1. Introduction. Our aim is to test numerically the new method of interpolation determinants (cf. [2], [6]) in the context of linear forms in two logarithms. In the recent years, M. Mignotte and M. Waldschmidt have used Schneider's construction in a series of papers [3]-[5] to get lower bounds for such a linear form with rational integer coefficients. They got relatively precise results with a numerical constant around a few hundreds. Here we take up Schneider's method again in the framework of interpolation determinants. We decrease the constant to less than one hundred when the logarithms involved are real numbers. Theorems 1 and 2 are simple corollaries of our main result which is Theorem 3. At first glance, the statement of Theorem 3 seems to be complicated, but it is much more precise than the above mentioned corollaries, which are only examples of applications. Let us also mention that we have been led in Section 3 to some technical lemmas which may be useful in some other situations apart from transcendental number theory.
A preliminary version of this text can also be found in [6], in the form of an appendix to lectures given by M. Waldschmidt at Madras Math. Science Institute. I would like to thank Dong Ping Ping and M. Waldschmidt for useful comments and remarks during the writing of this paper.
},

author = {Michel Laurent},

journal = {Acta Arithmetica},

keywords = {linear forms in logarithms; height; lower bound; linear combination of two logarithms of real algebraic numbers with integer coefficients; interpolation determinants},

language = {eng},

number = {2},

pages = {181-199},

title = {Linear forms in two logarithms and interpolation determinants},

url = {http://eudml.org/doc/206599},

volume = {66},

year = {1994},

}

TY - JOUR

AU - Michel Laurent

TI - Linear forms in two logarithms and interpolation determinants

JO - Acta Arithmetica

PY - 1994

VL - 66

IS - 2

SP - 181

EP - 199

AB -
1. Introduction. Our aim is to test numerically the new method of interpolation determinants (cf. [2], [6]) in the context of linear forms in two logarithms. In the recent years, M. Mignotte and M. Waldschmidt have used Schneider's construction in a series of papers [3]-[5] to get lower bounds for such a linear form with rational integer coefficients. They got relatively precise results with a numerical constant around a few hundreds. Here we take up Schneider's method again in the framework of interpolation determinants. We decrease the constant to less than one hundred when the logarithms involved are real numbers. Theorems 1 and 2 are simple corollaries of our main result which is Theorem 3. At first glance, the statement of Theorem 3 seems to be complicated, but it is much more precise than the above mentioned corollaries, which are only examples of applications. Let us also mention that we have been led in Section 3 to some technical lemmas which may be useful in some other situations apart from transcendental number theory.
A preliminary version of this text can also be found in [6], in the form of an appendix to lectures given by M. Waldschmidt at Madras Math. Science Institute. I would like to thank Dong Ping Ping and M. Waldschmidt for useful comments and remarks during the writing of this paper.

LA - eng

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

UR - http://eudml.org/doc/206599

ER -

## References

top- [1] J. Dieudonné, Calcul infinitésimal, Collection méthodes, Paris, 1968. Zbl0155.10001
- [2] M. Laurent, Sur quelques résultats récents de transcendance, Astérisque 198-200 (1991), 209-230. Zbl0762.11027
- [3] M. Mignotte and M. Waldschmidt, Linear forms in two logarithms and Schneider's method, Math. Ann. 231 (1978), 241-267. Zbl0349.10029
- [4] M. Mignotte and M. Waldschmidt, Linear forms in two logarithms and Schneider's method II, Acta Arith. 53 (1989), 251-287. Zbl0642.10034
- [5] M. Mignotte and M. Waldschmidt, Linear forms in two logarithms and Schneider's method III, Ann. Fac. Sci. Toulouse Math. 97 (1989), 43-75. Zbl0702.11044
- [6] M. Waldschmidt, Linear independence of logarithms of algebraic numbers, IMSc. Report No. 116, The Institute of Mathematical Sciences, Madras, 1992

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