Algebraic independence of elementary functions and its application to Masser's vanishing theorem

Abstract

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 = (ωij) be a nonsingular matrix inMn (ℤ) with no roots of unity as eigenvalue. LetP(z) be a nonzero polynomial inC[z],z = (z1,⋯,zn). Letx = (x1,⋯,xn) be an element ofCn withxi ≠ 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(Ωkx) = 0 for infinitely many positive integersk, thenx1,⋯,xn are multiplicatively dependent.