|
Criteria of algebraic independence with multiplicities and interpolation determinants
Author(s):
Michel
Laurent;
Damien
Roy
Journal:
Trans. Amer. Math. Soc.
351
(1999),
1845-1870.
MSC (1991):
Primary 11J85;
Secondary 11J04
Posted:
January 19, 1999
Retrieve article in:
PDF
This article is available free of charge
Abstract |
References |
Similar articles |
Additional information
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.
References:
- 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
Similar Articles:
Retrieve articles in Transactions of the American Mathematical Society
with MSC
(1991):
11J85,
11J04
Retrieve articles in all Journals with MSC
(1991):
11J85,
11J04
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:
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
Copyright of article:
Copyright
1999,
American Mathematical Society
|
Comments: webmaster@ams.org