On polynomials in additive and multiplicative functions

Abstract

In the theory of linear differential systems with constant coefficients ([2], 3.14), in the investigation of the partial differential equation of Peschl-Bauer, ([5], § 3), and in the theory of analytic iterations in rings of formal power series, theorems of the following type are useful:

• (A) Let λ ₁,...,λ _n be n distinct complex numbers and let P(X,Y ₁,...,Y _n ) be a polynomial in X and Y ₁,...,Y _n . Suppose that P(t,e ^{λ 1t},...,e^λnt)=0 for all t∈ ℂ. Then we have P=0.

• (B) Let λ ₁,...,λ _n be complex numbers such that for all (α ₁,...,α _n)∈ ℕ₀ with (α ₁,...,α _n)≠(0,...,0),$\sum\limits_{{i = 1}} {{\alpha_i}{\lambda_i}} = 0$ always holds. Moreover, let P(X, Y ₁,..,Y _n ) be a polynomial with complex coefficients which vanishes if we substitute X=t,Y ₁=e ^λ1t, t∈ ℂ Then we have again P=0.