Polynomial Equations for Additive Functions I: The Inner Parameter Case

Gselmann, Eszter and Kiss, Gergely (2024) Polynomial Equations for Additive Functions I: The Inner Parameter Case. RESULTS IN MATHEMATICS, 79 (2). No.-63. ISSN 1422-6383 (print); 1420-9012 (online)

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Abstract

The aim of this sequence of work is to investigate polynomial equations satisfied by additive functions. As a result of this, new characterization theorems for homomorphisms and derivations can be given. More exactly, in this paper the following type of equation is considered \begin{aligned} \sum _{i=1}^{n}f_{i}(x^{p_{i}})g_{i}(x^{q_{i}})= 0 \qquad \left( x\in \mathbb {F}\right) , \end{aligned} ∑ i = 1 n f i ( x p i ) g i ( x q i ) = 0 x ∈ F , where n is a positive integer, \mathbb {F}\subset \mathbb {C} F ⊂ C is a field, f_{i}, g_{i}:\mathbb {F}\rightarrow \mathbb {C} f i , g i : F → C are additive functions and p_i, q_i p i , q i are positive integers for all i=1, \ldots , n i = 1 , …,n .

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