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Criteria of algebraic independence with
multiplicities and interpolation determinants
Authors:
Michel Laurent and Damien Roy
Journal:
Trans. Amer. Math. Soc. 351 (1999), 1845-1870
MSC (1991):
Primary 11J85; Secondary 11J04
Posted:
January 19, 1999
MathSciNet review:
1475689
Full-text PDF Free Access
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Abstract: We generalize Gel'fond's criterion of algebraic independence by taking into account the values of the derivatives of the polynomials, and show how the new criterion applies to proving results of algebraic independence using interpolation determinants. We also establish a new result of approximation of a transcendental number by algebraic numbers of bounded degree and size. It contains an earlier result of E. Wirsing and also a result announced by A. Durand.
- 1.
W.
Dale Brownawell, Sequences of Diophantine approximations, J.
Number Theory 6 (1974), 11–21. MR 0337803
(49 #2572)
- 2.
Alain
Durand, Approximations algébriques d’un nombre
transcendant, Cinquante ans de polynômes (Paris, 1988) Lecture
Notes in Math., vol. 1415, Springer, Berlin, 1990,
pp. 94–96 (French). MR 1044106
(91b:11082), http://dx.doi.org/10.1007/BFb0084880
- 3.
K. Geddes, S. Czapor and G. Labahn, Algorithms for Computer Algebra, Kluwer Academic Publishers, 1992, pp. 285-300. MR 96a:63049
- 4.
Michel
Laurent, Sur quelques résultats récents de
transcendance, Astérisque 198-200 (1991),
209–230 (1992) (French). Journées Arithmétiques, 1989
(Luminy, 1989). MR 1144324
(93b:11091)
- 5.
Michel
Laurent, Linear forms in two logarithms and interpolation
determinants, Acta Arith. 66 (1994), no. 2,
181–199. MR 1276987
(95e:11083)
- 6.
Patrice
Philippon, Lemmes de zéros dans les groupes
algébriques commutatifs, Bull. Soc. Math. France
114 (1986), no. 3, 355–383 (French, with
English summary). MR 878242
(89c:11111)
- 7.
Damien
Roy and Michel
Waldschmidt, Quadratic relations between logarithms of algebraic
numbers, Proc. Japan Acad. Ser. A Math. Sci. 71
(1995), no. 7, 151–153. MR 1363903
(96k:11092)
- 8.
D. Roy et M. Waldschmidt, Approximation diophantienne et indépendance algébrique de logarithmes, Ann. Sci. École Norm. Sup. 30 (1997), 753-796. CMP 98:03
- 9.
R.
Tijdeman, On the algebraic independence of certain numbers,
Nederl. Akad. Wetensch. Proc. Ser. A 74=Indag. Math. 33
(1971), 146–162. MR 0294264
(45 #3333)
- 10.
M. Waldschmidt, Linear independence of logarithms of algebraic numbers, The Institute of Mathematical Sciences, Madras, IMSc. Report 116, 1992.
- 11.
Michel
Waldschmidt, Nombres transcendants, Lecture Notes in
Mathematics, Vol. 402, Springer-Verlag, Berlin, 1974 (French). MR 0360483
(50 #12931)
- 12.
Michel
Waldschmidt, Suites colorées d’après G. V.
Čudnovs′kiĭ, Séminaire
Delange-Pisot-Poitou, 17e année (1975/76), Théorie des
nombres: Fasc. 2, Exp. No. G21, Secrétariat Math., Paris, 1977,
pp. 11 (French). MR 0480373
(58 #540)
- 13.
Eduard
Wirsing, Approximation mit algebraischen Zahlen beschränkten
Grades, J. Reine Angew. Math. 206 (1960), 67–77
(German). MR
0142510 (26 #79)
- 1.
- W. D. Brownawell, Sequences of Diophantine approximations, J. Number Theory 6 (1974), 10-21. MR 49:2572
- 2.
- A. Durand, Approximations algébriques d'un nombre transcendant, Cinquante ans de polynômes, Lecture Notes in Math. 1415 (M. Langevin and M. Waldschmidt, eds.), Springer-Verlag, 1990, pp. 94-96. MR 91b:11082
- 3.
- K. Geddes, S. Czapor and G. Labahn, Algorithms for Computer Algebra, Kluwer Academic Publishers, 1992, pp. 285-300. MR 96a:63049
- 4.
- M. Laurent, Sur quelques résultats récents de transcendance, Journées Arithmétiques de Luminy 1989, Astérisque 198-200, pp. 209-230. MR 93b:11091
- 5.
- M. Laurent, Linear forms in two logarithms and interpolation determinants, Acta Arithmetica LXVI.2 (1994), 181-199. MR 95e:11083
- 6.
- P. Philippon, Lemme de zéros dans les groupes algébriques commutatifs, Bull. Soc. Math. France 114 (1986), 355-383, and 115 (1987), 397-398. MR 89c:11111
- 7.
- D. Roy and M. Waldschmidt, Quadratic relations between logarithms of algebraic numbers, Proc. Japan Acad. Ser. A Math. Sci. 71 (1995), 151-153. MR 96k:11092
- 8.
- D. Roy et M. Waldschmidt, Approximation diophantienne et indépendance algébrique de logarithmes, Ann. Sci. École Norm. Sup. 30 (1997), 753-796. CMP 98:03
- 9.
- R. Tijdeman, On the algebraic independence of certain numbers, Indag. Math. 33 (1971), 146-162. MR 45:3333
- 10.
- M. Waldschmidt, Linear independence of logarithms of algebraic numbers, The Institute of Mathematical Sciences, Madras, IMSc. Report 116, 1992.
- 11.
- M. Waldschmidt, Nombres transcendants, Lecture Notes in Math. 402, Springer-Verlag, 1974. MR 50:12931
- 12.
- M. Waldschmidt, Suites colorées, Séminaire Delange-Pisot-Poitou, (Groupe d'étude de théorie des nombres), 17-ième année, 1975/76, exposé 21, 11 pages. MR 58:540
- 13.
- E. Wirsing, Approximation mit algebraischen Zahlen beschränkten Grades, J. reine angew. Math. 206 (1961), 67-77. MR 26:79
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Additional Information
Michel Laurent
Affiliation:
Institut de Mathématiques de Luminy, CNRS, 163 Avenue de Luminy, Case 907, 13288 Marseille Cédex 9, France
Email:
laurent@iml.univ-mrs.fr
Damien Roy
Affiliation:
Département de Mathématiques, Université d’Ottawa, 585 King Edward, Ottawa, Ontario, Canada K1N 6N5
Email:
droy@mathstat.uottawa.ca
DOI:
http://dx.doi.org/10.1090/S0002-9947-99-02216-3
PII:
S 0002-9947(99)02216-3
Received by editor(s):
March 12, 1997
Posted:
January 19, 1999
Additional Notes:
Second author partially supported by NSERC and CICMA
Article copyright:
© Copyright 1999 American Mathematical Society
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