Monomial functions, normal polynomials and polynomial equations

TL;DR

The paper investigates generalized monomial functions $f: \mathbb{F} \to \mathbb{C}$ (of degree $n$) under the constraint that $x \mapsto f(P(x))$ is a (normal) polynomial, and develops a reduction to monomial cases using polarization and multi-additive representations. It proves that if $x \mapsto f(x^k)$ is a normal polynomial then $f$ has degree $kn$ and can be expressed via additive homomorphisms, enabling a precise structural description. The main result characterizes solutions by showing there exist field homomorphisms $\varphi_1,\varphi_2$ such that $f(x)=f(1)\varphi_1(x)\varphi_2(x)$ and the associated additive components satisfy $a_i(x)=a_i(1)\varphi_i(x)$, thereby identifying generalized monomials compatible with polynomial equations of the form $f(P(x))=Q(g(x))$. This provides a rigid classification of such functional equations over fields of characteristic zero and clarifies how generalized polynomial frameworks interact with classical polynomial constraints.

Abstract

In this paper we consider generalized monomial functions $f, g\colon \mathbb{F}\to \mathbb{C}$ (of possibly different degree) that also fulfill \[ f(P(x))= Q(g(x)) \qquad \left(x\in \mathbb{F}\right), \] where $P\in \mathbb{F}[x]$ and $Q\in \mathbb{C}[x]$ are given (classical) polynomials.

Theorems & Definitions (39)

• Theorem 1
• Definition 1
• Theorem 2: Polarization formula
• Corollary 1
• Lemma 1
• Definition 2
• Remark 1
• Definition 3
• Remark 2
• Remark 3