A note on a conjecture by Ulas on polynomial substitutions

TL;DR

This paper proves a recent conjecture by Ulas concerning the reducibility of polynomial substitutions, contributing to the understanding of polynomial structure and substitution properties.

Contribution

It provides a proof for Ulas's conjecture on reducible polynomial substitutions, advancing theoretical knowledge in polynomial algebra.

Findings

Confirmed the conjecture's validity

Enhanced understanding of polynomial substitution reducibility

Contributed to algebraic theory of polynomial structures

Abstract

We prove a recent conjecture by Ulas on reducible polynomial substitutions.

Full text

We prove the following result, which was conjectured by Ulas in [2] and proven for $d\leq 4$ there.

We obtain $f(h(\frac{1}{\alpha}))=f(\alpha)=0$. So $f(h(X))$ shares the root $\frac{1}{\alpha}$ with the irreducible polynomial $g(X)$, thus $g(X)$ divides $f(h(X))$. The degree of $h$ is at least $2$, for otherwise $\alpha$ would have degree at most $2$ over $K$, contrary to $d\geq 3$.

The assertion now follows from $\deg f(h(X))\geq 2d$ and $\deg g(X)=d$. ∎

(b) The answer to [2, Question 5.5], a kind of converse to the above theorem, is negative if $K=\mathbb{R}$ by lack of irreducible polynomials of degree $\geq 3$.