Abstract:
A theorem on the successive minima of lattices corresponding to the integer solutions of systems of linear equations is proved. As a corollary, theorems on the successive minima are obtained for the set of solutions of equations of the form
x1lnα1+⋯+xnlnαn=lnβ,x1,…,xn∈Z,
for fixed α1,…,αn in an algebraic number field K and for variable β∈K equal either to 1 or a root of unity.
\Bibitem{Mat93}
\by E.~M.~Matveev
\paper On linear and multiplicative relations
\jour Russian Acad. Sci. Sb. Math.
\yr 1994
\vol 78
\issue 2
\pages 411--425
\mathnet{http://mi.mathnet.ru/eng/sm977}
\crossref{https://doi.org/10.1070/SM1994v078n02ABEH003477}
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\zmath{https://zbmath.org/?q=an:0816.11037}
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Linking options:
https://www.mathnet.ru/eng/sm977
https://doi.org/10.1070/SM1994v078n02ABEH003477
https://www.mathnet.ru/eng/sm/v184/i4/p23
This publication is cited in the following 13 articles:
Tao Zheng, Lecture Notes in Computer Science, 12291, Computer Algebra in Scientific Computing, 2020, 621
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
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W. A. Coppel, Number Theory, 2009, 327
A. Dubickas, “Multiplicative relations with conjugate algebraic numbers”, Ukr Math J, 59:7 (2007), 984
Number Theory, 2006, 385
E. M. Matveev, “The index of multiplicative groups of algebraic numbers”, Sb. Math., 196:9 (2005), 1307–1318
Loher T., Masser D., “Uniformly Counting Points of Bounded Height”, Acta Arith., 111:3 (2004), 277–297
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
E. M. Matveev, “An explicit lower bound for a homogeneous rational linear form in logarithms of algebraic numbers”, Izv. Math., 62:4 (1998), 723–772
Daniel Bertrand, “Duality on tori and multiplicative dependence relations”, J Austral Math Soc, 62:2 (1997), 198
Bertrand D., “Minimal Heights and Polarizations on Group Varieties”, Duke Math. J., 80:1 (1995), 223–250
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