Two questions on polynomial decomposition

TL;DR

This paper investigates conditions under which a univariate polynomial over a ring can be decomposed into compositions of lower-degree polynomials, addressing questions about the existence of such decompositions over rings and their extensions.

Contribution

It provides answers to Gusic's questions on polynomial decomposition over rings and introduces new related questions for further research.

Findings

Conditions for polynomial decomposition over rings and extensions

Implications of existence of decompositions over extensions

New open questions on polynomial composition

Abstract

Given a univariate polynomial f(x) over a ring R, we examine when we can write f(x) as g(h(x)) where g and h are polynomials of degree at least 2. We answer two questions of Gusic regarding when the existence of such g and h over an extension of R implies the existence of such g and h over R. We also pose two new questions along these lines.