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Quadratic functions as solutions of polynomial equations

Eszter Gselmann, Mehak Iqbal

Abstract

The so-called polynomial equations play an important role both in algebra and in the theory of functional equations. If the unknown functions in the equation are additive, relatively many results are known. However, even in this case, there are a lot of open questions. In some specific cases, according to classical results, the unknown additive functions are homomorphisms, derivations, or linear combinations of these. The question arises as to whether the solutions can be described even if the unknown functions are not assumed to be additive but to be generalized monomials. As a starting point, we will deal with quadratic functions in this paper. We aim to show that quadratic functions that are solutions to certain polynomial equations necessarily have a `special' form. Further, we also present a method to determine these special forms.

Quadratic functions as solutions of polynomial equations

Abstract

The so-called polynomial equations play an important role both in algebra and in the theory of functional equations. If the unknown functions in the equation are additive, relatively many results are known. However, even in this case, there are a lot of open questions. In some specific cases, according to classical results, the unknown additive functions are homomorphisms, derivations, or linear combinations of these. The question arises as to whether the solutions can be described even if the unknown functions are not assumed to be additive but to be generalized monomials. As a starting point, we will deal with quadratic functions in this paper. We aim to show that quadratic functions that are solutions to certain polynomial equations necessarily have a `special' form. Further, we also present a method to determine these special forms.
Paper Structure (2 sections, 13 theorems, 144 equations)

This paper contains 2 sections, 13 theorems, 144 equations.

Table of Contents

  1. Introduction and preliminaries
  2. Results

Key Result

Theorem 1

Let $(\mathbb{K}, +,\cdot)$ be a field of characteristic zero, let $(\mathbb{F}, +,\cdot)$ be a subfield of $(\mathbb{K}, +,\cdot)$, let $S$ be an algebraic base of $\mathbb{K}$ over $\mathbb{F}$, if it exists, and let $S=\emptyset$ otherwise. Let $f\colon \mathbb{F}\to \mathbb{K}$ be a derivation.

Theorems & Definitions (34)

  • Theorem 1
  • Definition 1
  • Theorem 2: Polarization formula
  • Corollary 1
  • Lemma 1
  • Definition 2
  • Remark 1
  • Definition 3
  • Remark 2
  • Remark 3
  • ...and 24 more