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PARAMETRIC GEOMETRY OF NUMBERS IN FUNCTION FIELDS

Part of: Geometry of numbers Topological rings and modules Diophantine approximation, transcendental number theory Approximations and expansions

Published online by Cambridge University Press:  29 November 2017

Damien Roy  and
Michel Waldschmidt
Damien Roy
Affiliation:
Département de mathématiques et de statistique, Université d’Ottawa, 585, Avenue King Edward, Ottawa, Ontario, Canada K1N 6N5 email [email protected]
Michel Waldschmidt
Affiliation:
Sorbonne Universités, UPMC Univ Paris 06, UMR 7586 IMJ-PRG, F - 75005 Paris, France email [email protected]
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Abstract

We transpose the parametric geometry of numbers, recently created by Schmidt and Summerer, to fields of rational functions in one variable and analyze, in that context, the problem of simultaneous approximation to exponential functions.

MSC classification

Primary: 11J13: Simultaneous homogeneous approximation, linear forms
Secondary: 41A20: Approximation by rational functions 41A21: Padé approximation 13J05: Power series rings 11H06: Lattices and convex bodies
Type
Research Article
Information
Mathematika , Volume 63 , Issue 3 , 2017 , pp. 1114 - 1135
Copyright
Copyright © University College London 2017 

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References

Baker, A., On an analogue of Littlewood’s Diophantine approximation problem. Michigan Math. J. 11 1964, 247–250.Google Scholar
Hermite, Ch., Sur la fonction exponentielle. C. R. Acad. Sci. Paris 77 1873, 18–24 74–79, 226–233, 285–293; Œuvres de Charles Hermite, Vol. 3, 150–181.Google Scholar
Hermite, Ch., Sur la généralisation des fractions continues algébriques (extrait d’une lettre à M Pincherle). Ann. Mat. 21 1893, 289–308; Œuvres de Charles Hermite, Vol. 4, 357–377.Google Scholar
Jager, H., A multidimensional generalization of the Padé table I–VI. Nederl. Akad. Wet. Proc. Ser. A 67 1964, 193–249.Google Scholar
Keita, A., Continued fractions and parametric geometry of numbers. J. Théor. Nombres Bordeaux 29 2017, 129–135.Google Scholar
Mahler, K., Zur Approximation der Exponentialfunktion und des Logarithmus I. J. Reine Angew. Math. 166 1931, 118–136.Google Scholar
Mahler, K., An analogue to Minkowski’s geometry of numbers in a field of series. Ann. of Math. (2) 42 1941, 488–522.Google Scholar
Mahler, K., On compound convex bodies I, II. Proc. Lond. Math. Soc. (3) 5 1955, 358–384.Google Scholar
Mahler, K., Perfect systems. Compos. Math. 19 1968, 95–166.Google Scholar
Roy, D., On Schmidt and Summerer parametric geometry of numbers. Ann. of Math. (2) 182 2015, 739–786.Google Scholar
Schmidt, W. M., Open problems in Diophantine approximations. In Approximations diophantiennes et nombres transcendants (Luminy, 1982) (Progress in Mathematics 31 ), Birkhäuser (Boston, MA, 1983), 271–287.Google Scholar
Schmidt, W. M. and Summerer, L., Parametric geometry of numbers and applications. Acta Arith. 140 2009, 67–91.Google Scholar
Schmidt, W. M. and Summerer, L., Diophantine approximation and parametric geometry of numbers. Monatsh. Math. 169 2013, 51–104.Google Scholar
Thunder, J. L., Siegel’s lemma for function fields. Michigan Math. J. 42 1995, 147–162.Google Scholar