Algebraic independence of elementary functions and its application to Masser's vanishing theorem
 Keiji Nishioka,
 Kumiko Nishioka
 … show all 2 hide
Rent the article at a discount
Rent now* Final gross prices may vary according to local VAT.
Get AccessSummary
Here is an improvement on Masser's Refined Identity (D. W. Masser:A vanishing theorem for power series. Invent. Math.67 (1982), 275–296). The present method depends on a result from differential algebra andpadic analysis. The investigation from the viewpoint ofpadic analysis makes the proof clearer and, in particular, it is possible to exclude the concept of “density” which is necessary in Masser's treatment. That is to say, the theorem will be stated as follows: Let Ω = (ω _{ ij }) be a nonsingular matrix inM _{ n } (ℤ) with no roots of unity as eigenvalue. LetP(z) be a nonzero polynomial inC[z],z = (z _{1},⋯,z _{ n }). Letx = (x _{1},⋯,x _{ n }) be an element ofC ^{ n } withx _{ i } ≠ 0 for eachi. Define $$\Omega x = \left( {\prod\limits_{i = 1}^n {x_i^{\omega 1_i } ,...,} \prod\limits_{i = 1}^n {x_i^{\omega _{ni} } } } \right)$$ . IfP(Ω^{ k } x) = 0 for infinitely many positive integersk, thenx _{1},⋯,x _{ n } are multiplicatively dependent.
To prove this, the following fact on elementary functions will be needed: LetK be an ordinary differential field andC be its field of constants. LetR be a differential field extension ofK andu _{1},⋯,u _{ m } be elements ofR such that the field of constants ofR is the same asC and for eachi the field extensionK _{ i } =K(u _{1},⋯,u _{ i }) ofK is a differential one such thatu′_{ i } =t′_{ i−1} u _{ i } for somet _{ i−1}∈K _{ i−1} oru _{ i } is algebraic overK _{ i−1}. Letf _{1},⋯,f _{ n } ∈R be distinct elements moduloC and suppose that for eachi there is a nonzeroe _{ i } ∈R withe′ _{ i } =f′ _{ i } e _{ i }. Thene _{1},⋯,e _{ n } are linearly independent overK.
 Ax, J.,On Schanuel's conjectures. Annals of Math.93 (1971), 252–268.
 Cassels, J. W. S.,An embedding theorem for fields. Bull. Austral. Math. Soc.14 (1976), 193–198.
 Koblitz, N.,padic numbers, padic analysis and zetafunctions (2nd ed. (Graduate Texts in Math. Vol. 58). Springer, New York and Berlin, 1984.
 Kolchin, E. R.,Algebraic groups and algebraic dependence. Amer. J. Math.90 (1968), 1151–1164.
 Masser, D. W.,A vanishing theorem for power series. Invent. Math.67 (1982), 275–296. CrossRef
 Rosenlicht, M.,On Liouville's theory of elementary functions. Pacific J. Math.65 (1976), 485–492.
 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.
 Title
 Algebraic independence of elementary functions and its application to Masser's vanishing theorem
 Journal

aequationes mathematicae
Volume 40, Issue 1 , pp 6777
 Cover Date
 19901201
 DOI
 10.1007/BF02112281
 Print ISSN
 00019054
 Online ISSN
 14208903
 Publisher
 BirkhäuserVerlag
 Additional Links
 Topics
 Keywords

 Primary 11J81
 Secondary 12H05
 Industry Sectors
 Authors

 Keiji Nishioka ^{(1)} ^{(2)}
 Kumiko Nishioka ^{(1)} ^{(2)}
 Author Affiliations

 1. Takabatakecho 184632, 630, Nara, Japan
 2. Department of Mathematics, Nara Women's University, KitaUoya Nishimachi, 630, Nara, Japan