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Izv. RAN. Ser. Mat., 1998, Volume 62, Issue 4, Pages 81–136 (Mi izv190)  

This article is cited in 25 scientific papers (total in 25 papers)

An explicit lower bound for a homogeneous rational linear form in logarithms of algebraic numbers

E. M. Matveev

Moscow State Textile Academy named after A. N. Kosygin

Abstract: In this paper we study linear forms $\Lambda=b_1\ln\alpha_1+…+b_n\ln\alpha_n$ with rational integer coefficients $b_j$ ($b_n\ne 0$, $n\geqslant 2$), where the $\alpha_j$ are algebraic numbers satisfying the so-called strong independence condition. In standard notation, we prove an explicit estimate of the form
$$ |\Lambda|>\exp(-C^nD^{n+2}\Omega\ln(C^nD^{n+2}\Omega')\ln(eB)). $$
Its novel feature is that it contains no factors of the form $n^n$.

DOI: https://doi.org/10.4213/im190

Full text: PDF file (3445 kB)
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English version:
Izvestiya: Mathematics, 1998, 62:4, 723–772

Bibliographic databases:

MSC: 11J86, 11J25
Received: 24.07.1996

Citation: E. M. Matveev, “An explicit lower bound for a homogeneous rational linear form in logarithms of algebraic numbers”, Izv. RAN. Ser. Mat., 62:4 (1998), 81–136; Izv. Math., 62:4 (1998), 723–772

Citation in format AMSBIB
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\by E.~M.~Matveev
\paper An explicit lower bound for a~homogeneous rational linear form in logarithms of algebraic numbers
\jour Izv. RAN. Ser. Mat.
\yr 1998
\vol 62
\issue 4
\pages 81--136
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    1. Yu K.R., “p-adic logarithmic forms and group varieties II”, Acta Arithmetica, 89:4 (1999), 337–378  crossref  mathscinet  zmath  isi  scopus
    2. E. M. Matveev, “An explicit lower bound for a homogeneous rational linear form in the logarithms of algebraic numbers. II”, Izv. Math., 64:6 (2000), 1217–1269  mathnet  crossref  crossref  mathscinet  zmath  isi
    3. Waldschmidt M., “On a problem of Mahler concerning the approximation of exponentials and logarithms”, Publicationes Mathematicae-Debrecen, 56:3–4 (2000), 713–738  mathscinet  zmath  isi
    4. Stewart C.L., Yu K., “On the abc conjecture, II”, Duke Mathematical Journal, 108:1 (2001), 169–181  crossref  mathscinet  zmath  isi  scopus  scopus
    5. Okazaki R., “Geometry of a cubic Thue equation”, Publicationes Mathematicae-Debrecen, 61:3–4 (2002), 267–314  mathscinet  zmath  isi
    6. Bugeaud Y., “Linear forms in two m-adic logarithms and applications to Diophantine problems”, Compositio Mathematica, 132:2 (2002), 137–158  crossref  mathscinet  zmath  isi  elib  scopus  scopus
    7. Gyory K., “Solving diophantine equations by Baker's theory”, Panorama in Number Theory Or the View From Baker'S Garden, 2002, 38–72  mathscinet  zmath  isi
    8. Yu K.R., “Report on p-adic logarithmic forms”, Panorama in Number Theory Or the View From Baker'S Garden, 2002, 11–25  mathscinet  isi
    9. Bilu Y.F., “Baker's method and modular curves”, Panorama in Number Theory Or the View From Baker'S Garden, 2002, 73–88  mathscinet  zmath  isi
    10. Nesterenko Yu., “Linear forms in logarithms of rational numbers”, Diophantine approximation (Cetraro, 2000), Lecture Notes in Math., 1819, Springer, Berlin, 2003, 53–106  crossref  mathscinet  zmath  isi
    11. Waldschmidt M., “Linear independence measures for logarithms of algebraic numbers”, Diophantine approximation (Cetraro, 2000), Lecture Notes in Math., 1819, Springer, Berlin, 2003, 249–344  crossref  mathscinet  zmath  isi
    12. Bilu Yu.F., “Catalan's conjecture (after Mihǎilescu)”, Astérisque, 294, 2004, 1–26  mathscinet  zmath  isi
    13. E. M. Matveev, “The index of multiplicative groups of algebraic numbers”, Sb. Math., 196:9 (2005), 1307–1318  mathnet  crossref  crossref  mathscinet  zmath  isi  elib
    14. Simons J., De W.eger B., “Theoretical and computational bounds for m-cycles of the 3n+1-problem”, Acta Arithmetica, 117:1 (2005), 51–70  crossref  mathscinet  zmath  adsnasa  isi  elib  scopus  scopus
    15. Yu. M. Alexencev, “Index of Lattices and Hilbert Polynomials”, Math. Notes, 80:3 (2006), 313–317  mathnet  crossref  crossref  mathscinet  zmath  isi
    16. Yu. M. Aleksentsev, “The Hilbert polynomial and linear forms in the logarithms of algebraic numbers”, Izv. Math., 72:6 (2008), 1063–1110  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib  elib
    17. Akhtari, S, “Cubic Thue equations”, Publicationes Mathematicae-Debrecen, 75:3–4 (2009), 459  mathscinet  zmath  isi
    18. Akhtari, S, “Quartic Thue equations”, Journal of Number Theory, 130:1 (2010), 40  crossref  mathscinet  zmath  isi  scopus
    19. Akhtari Sh., “Representation of Unity by Binary Forms”, Trans. Am. Math. Soc., 364:4 (2012), 2129–2155  crossref  mathscinet  zmath  isi  elib  scopus
    20. Yu K., “P-Adic Logarithmic Forms and a Problem of Erdas”, Acta Math., 211:2 (2013), 315–382  crossref  mathscinet  zmath  isi  scopus
    21. Kim J. Stewart C.L., “Well Spaced Integers Generated By An Infinite Set of Primes”, Proc. Amer. Math. Soc., 143:3 (2015), 915–923  crossref  mathscinet  zmath  isi
    22. Koymans P.H., “The Catalan Equation”, Indag. Math.-New Ser., 28:2 (2017), 321–352  crossref  mathscinet  zmath  isi  scopus
    23. Bugeaud Y., “Linear Forms in Logarithms and Applications”, Linear Forms in Logarithms and Applications, Irma Lectures in Mathematics and Theoretical Physics, 28, Eur. Math. Soc., 2018, 1–224  crossref  mathscinet  isi
    24. Stewart (Waterloo) C. L., “Sets Generated By Finite Sets of Algebraic Numbers”, Acta Arith., 184:2 (2018), 193–200  crossref  mathscinet  zmath  isi  scopus
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