On the equation $f(g(x)) =f(x)h^m(x)$ for composite polynomials

TL;DR

This paper characterizes solutions to the polynomial functional equation $f(g(x))=f(x)h^m(x)$ under specific conditions, showing no solutions for degree ≥ 3 and explicitly solving the quadratic case using Chebyshev polynomials, with applications to number theory.

Contribution

It provides a complete classification of solutions to the equation for polynomials over arbitrary fields, especially identifying the quadratic case solutions explicitly.

Findings

No solutions for $ ext{deg} f extgreater 2$.

Explicit solutions for $ ext{deg} f = 2$ using Chebyshev polynomials.

Applications to Diophantine problems and conjectures in number theory.

Abstract

In this paper we solve the equation $f(g(x))=f(x)h^m(x)$ where $f(x)$, $g(x)$ and $h(x)$ are unknown polynomials with coefficients in an arbitrary field $K$, $f(x)$ is non-constant and separable, $\deg g \geq 2$, the polynomial $g(x)$ has non-zero derivative $g'(x) \ne 0$ in $K[x]$ and the integer $m \geq 2$ is not divisible by the characteristic of the field $K$. We prove that this equation has no solutions if $\deg f \geq 3$. If $\deg f = 2$, we prove that $m = 2$ and give all solutions explicitly in terms of Chebyshev polynomials. The diophantine applications for such polynomials $f(x)$, $g(x)$, $h(x)$ with coefficients in $\Q$ or $\Z$ are considered in the context of the conjecture of Cassaign et. al on the values of Louiville's $\lambda$ function at points $f(r)$, $r \in \Q$.