Nishioka, K. & Nishioka, K. Aeq. Math. (1990) 40: 67. doi:10.1007/BF02112281
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
. IfP(Ωkx) = 0 for infinitely many positive integersk, thenx1,⋯,xn 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 andu1,⋯,um be elements ofR such that the field of constants ofR is the same asC and for eachi the field extensionKi =K(u1,⋯,ui) ofK is a differential one such thatu′i =t′i−1ui for someti−1∈Ki−1 orui is algebraic overKi−1. Letf1,⋯,fn ∈R be distinct elements moduloC and suppose that for eachi there is a nonzeroei ∈R withe′i =f′iei. Thene1,⋯,en are linearly independent overK.