Polynomial identities satisfied by generalized polynomials

TL;DR

This paper characterizes degree-two generalized polynomials over fields satisfying specific polynomial equations, exploring the differences between generalized and normal polynomials and their polynomial identities.

Contribution

It provides new characterization theorems for degree-two generalized polynomials satisfying polynomial equations, clarifying the relationship between generalized and normal polynomials.

Findings

Characterization theorems for degree-two generalized polynomials

Analysis of the connection between generalized and normal polynomials

Insights into polynomial identities satisfied by generalized polynomials

Abstract

The main purpose of this paper is solve polynomial equations that are satisfied by (generalized) polynomials. More exactly, we deal with the following problem: let $\mathbb{F}$ be a field with $\mathrm{char}(\mathbb{F})=0$ and $P\in \mathbb{F}[x]$ and $Q\in \mathbb{C}[x]$ be polynomials. Our aim is to prove characterization theorems for generalized polynomials $f\colon \mathbb{F}\to \mathbb{C}$ of degree two that also fulfill equation \[ f(P(x))= Q(f(x)) \] for each $x\in \mathbb{F}$. As it turns out, the difficulty of such problems heavily depends on that we consider the above equation for generalized polynomials or for (normal) polynomials. Therefore, firstly we study the connection between these two notions.