×

Algebraic independence of elementary functions and its application to Masser’s vanishing theorem. (English) Zbl 0723.11033

Soit \(\Omega =(\omega_{ij})\) une matrice \(n\times n\) à coefficients dans \({\mathbb{Z}}\). On suppose que \(\Omega\) est inversible et que ses valeurs propres ne sont pas racines de l’unité. Si C est un corps de caractéristique nulle et \(x=(x_ 1,...,x_ n)\in C^{\times n}\), on note \[ \Omega x=(\prod_{1\leq i\leq n}x_ i^{\omega_{ji}})_{1\leq j\leq n}\in C^{\times n}. \] Les AA. montrent que, si \(P\in C[X]\setminus \{0\}\) et \(P(\Omega^ kx)=0\) pour une infinité d’entiers \(k\geq 0\), alors \(x_ 1,...,x_ n\) sont multiplicativement dépendants. Ils en déduisent la “refined identity” de D. W. Masser [Invent. Math. 67, 275-296 (1982; Zbl 0481.10034)] qui permet de montrer le lemme de zéros utile pour la méthode de transcendance de K. Mahler. Alors que D. W. Masser utilisait les résultats de J. Ax [Ann. Math., II. Ser. 93, 252-268 (1971; Zbl 0232.10026)] et de l’analyse complexe, le procédé des AA. paraît plus naturel: ils utilisent des résultats d’algèbre différentielle à la M. Rosenlicht [Pac. J. Math. 65, 485-492 (1976; Zbl 0318.12107)] et plongent la situation dans un \({\mathbb{C}}_ p\) convenable [suivant J. W. S. Cassels, Bull. Aust. Math. Soc. 14, 193-198 (1976; Zbl 0345.12104)].

MSC:

11J85 Algebraic independence; Gel’fond’s method
11J81 Transcendence (general theory)
12H05 Differential algebra
11S80 Other analytic theory (analogues of beta and gamma functions, \(p\)-adic integration, etc.)
PDFBibTeX XMLCite
Full Text: DOI EuDML

References:

[1] Ax, J.,On Schanuel’s conjectures. Annals of Math.93 (1971), 252–268. · Zbl 0232.10026 · doi:10.2307/1970774
[2] Cassels, J. W. S.,An embedding theorem for fields. Bull. Austral. Math. Soc.14 (1976), 193–198. · Zbl 0345.12104 · doi:10.1017/S000497270002503X
[3] Koblitz, N.,p-adic numbers, p-adic analysis and zeta-functions (2nd ed. (Graduate Texts in Math. Vol. 58). Springer, New York and Berlin, 1984. · Zbl 0364.12015
[4] Kolchin, E. R.,Algebraic groups and algebraic dependence. Amer. J. Math.90 (1968), 1151–1164. · Zbl 0169.36701 · doi:10.2307/2373294
[5] Masser, D. W.,A vanishing theorem for power series. Invent. Math.67 (1982), 275–296. · Zbl 0489.10026 · doi:10.1007/BF01393819
[6] Rosenlicht, M.,On Liouville’s theory of elementary functions. Pacific J. Math.65 (1976), 485–492. · Zbl 0318.12107
[7] Rosenlicht, M. andSinger, M.,On elementary, generalized elementary and Liouvillian extension fields. InContributions to Algebra, Bass, Cassidy, Kovacic eds., Academic Press, New York, 1977, pp. 329–342.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.