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Effective approximation to complex algebraic numbers by quadratic numbers

Part of: Diophantine approximation, transcendental number theory

Published online by Cambridge University Press:  08 January 2025

Prajeet Bajpai  and
Yann Bugeaud Open the ORCID record for Yann Bugeaud [Opens in a new window]
Prajeet Bajpai
Affiliation:
Department of Mathematics, University of British Columbia, Vancouver, B.C., V6T 1Z2 Canada e-mail: [email protected]
Yann Bugeaud*
Affiliation:
I.R.M.A., UMR 7501, Université de Strasbourg et CNRS, 7 rue René Descartes, 67084 Strasbourg, France Institut universitaire de France
*
e-mail: [email protected]
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Abstract

We establish an effective improvement on the Liouville inequality for approximation to complex nonreal algebraic numbers by quadratic complex algebraic numbers.

Keywords

Approximation to algebraic numberslinear forms in logarithms

MSC classification

Primary: 11J68: Approximation to algebraic numbers
Secondary: 11J86: Linear forms in logarithms; Baker's method
Type
Article
Information
Canadian Mathematical Bulletin , Volume 68 , Issue 1 , March 2025 , pp. 262 - 269
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Copyright
© The Author(s), 2025. Published by Cambridge University Press on behalf of Canadian Mathematical Society

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References

Bajpai, P. and Bugeaud, Y., Effective approximation to complex algebraic numbers by algebraic numbers of bounded degree . Trans. Amer. Math. Soc. 377(2024), 5247–5269.Google Scholar
Baker, A., Linear forms in the logarithms of algebraic numbers I–IV . Mathematika 13(1966), 204–216; 14(1967), 102–107 and 220–224; 15(1968), 204–216.CrossRefGoogle Scholar
Baker, A., A sharpening of the bounds for linear forms in logarithms II . Acta Arith. 24(1973), 33–36.CrossRefGoogle Scholar
Bugeaud, Y., Linear forms in logarithms and applications, IRMA Lectures in Mathematics and Theoretical Physics, 28, European Mathematical Society, Zürich, 2018.CrossRefGoogle Scholar
Bugeaud, Y. and Evertse, J.-H., Approximation of complex algebraic numbers by algebraic numbers of bounded degree . Ann. Sc. Norm. Super. Pisa Cl. Sci. 8(2009), no. 5, 333–368.Google Scholar
Evertse, J. H. and Győry, K., Unit equations in Diophantine number theory, Cambridge Studies in Advanced Mathematics, 146, Cambridge University Press, Cambridge, 2015.CrossRefGoogle Scholar
Fel’dman, N. I., An effective refinement of the exponent in Liouville’s theorem . Iz. Akad. Nauk SSSR, Ser. Mat. 35(1971), 973–990 (in Russian); English translation in Math. USSR. Izv. 5 (1971) 985–1002.Google Scholar
Güting, R., Polynomials with multiple zeros . Mathematika 14(1967), 149–159.CrossRefGoogle Scholar
Loxton, J. H. and van der Poorten, A. J., Multiplicative dependence in number fields . Acta Arith. 42(1983), 291–302.CrossRefGoogle Scholar
Waldschmidt, M., Diophantine approximation on linear algebraic groups . In: Transcendence Properties of the Exponential Function in Several Variables, Grundlehren Math. Wiss., 326, Springer, Berlin, 2000, xxiv+633 pp.Google Scholar