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

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 and p-adic analysis. The investigation from the viewpoint of p-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 in M_n (ℤ) with no roots of unity as eigenvalue. Let P(z) be a nonzero polynomial in C[z], z = (z₁,⋯, z_n). Let x = (x₁,⋯, x_n) be an element of Cⁿ with x_i ≠ 0 for each i. Define {Mathematical expression}. If P(Ω^kx) = 0 for infinitely many positive integers k, then x₁,⋯, x_n are multiplicatively dependent. To prove this, the following fact on elementary functions will be needed: Let K be an ordinary differential field and C be its field of constants. Let R be a differential field extension of K and u₁,⋯, u_m be elements of R such that the field of constants of R is the same as C and for each i the field extension K_i =K(u₁,⋯, u_i) of K is a differential one such that u′_i =t′_i-1u_i for some t_i-1∈K_i-1 or u_i is algebraic over K_i-1. Let f₁,⋯, f_n ∈R be distinct elements modulo C and suppose that for each i there is a nonzero e_i ∈R with e′_i =f′_ie_i. Then e₁,⋯, e_n are linearly independent over K.