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A functional equation for monomial functions

Eszter Gselmann, Mehak Iqbal

Abstract

Let $\mathbb{F}\subset \mathbb{K}$ be fields with characteristic zero, $n$ be a positive integer and $κ\in \mathbb{K}$. In this paper, we determine those monomials $f\colon \mathbb{F}\to \mathbb{K}$ of degree $n$ for which \[ f(x^{2})= κ\cdot x^{n}f(x) \] holds for all $x\in \mathbb{F}$. We show that similar to the classical results, where additive functions were considered, the monomial functions in the equation can be represented with the aid of homomorphisms and higher-order derivations.

A functional equation for monomial functions

Abstract

Let be fields with characteristic zero, be a positive integer and . In this paper, we determine those monomials of degree for which holds for all . We show that similar to the classical results, where additive functions were considered, the monomial functions in the equation can be represented with the aid of homomorphisms and higher-order derivations.

Paper Structure

This paper contains 2 sections, 6 theorems, 92 equations.

Table of Contents

  1. Introduction
  2. Results

Key Result

Theorem 1

Suppose that $G$ is a commutative semigroup, $S$ is a commutative group, $n\in \mathbb{N}$. If $A\colon G^{n}\to S$ is a symmetric, $n$-additive function, then for all $x, y_{1}, \ldots, y_{m}\in G$ we have

Theorems & Definitions (14)

  • Definition 1
  • Theorem 1: Polarization formula
  • Corollary 1
  • Lemma 1
  • Definition 2
  • Remark 1
  • Definition 3
  • Theorem 2
  • proof
  • Remark 2
  • ...and 4 more