caa a22 4500 465825443 CHVBK 20180323112147.0 cr unu---uuuuu 170327e19901201xx s 000 0 eng 10.1007/BF02112281 doi (NATIONALLICENCE)springer-10.1007/BF02112281 Algebraic independence of elementary functions and its application to Masser's vanishing theorem [Elektronische Daten] [Keiji Nishioka, Kumiko Nishioka] Summary: 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 andp-adic analysis. The investigation from the viewpoint ofp-adic 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. Birkhäuser Verlag, 1990 Primary 11J81 nationallicence Secondary 12H05 nationallicence Nishioka Keiji Takabatake-cho 184-632, 630, Nara, Japan aut Nishioka Kumiko Takabatake-cho 184-632, 630, Nara, Japan aut aequationes mathematicae Birkhäuser-Verlag 40/1(1990-12-01), 67-77 0001-9054 40:1<67 1990 40 10 https://doi.org/10.1007/BF02112281 text/html Onlinezugriff via DOI 1 research-article jats NATIONALLICENCE 856 40 https://doi.org/10.1007/BF02112281 text/html Onlinezugriff via DOI NATIONALLICENCE 700 1- Nishioka Keiji Takabatake-cho 184-632, 630, Nara, Japan aut NATIONALLICENCE 700 1- Nishioka Kumiko Takabatake-cho 184-632, 630, Nara, Japan aut NATIONALLICENCE 773 0- aequationes mathematicae Birkhäuser-Verlag 40/1(1990-12-01), 67-77 0001-9054 40:1<67 1990 40 10 Metadata rights reserved Springer special CC-BY-NC licence nationallicence BK010053 XK010053 XK010000 NATIONALLICENCE NATIONALLICENCE NL-springer