Symmetries of various sets of polynomials

TL;DR

This paper characterizes all linear bijections of polynomial rings over characteristic zero fields that preserve certain special sets of polynomials, showing they are essentially affine transformations scaled by constants.

Contribution

It establishes that such preservers are precisely affine transformations composed with scalar multiplication, extending known results to new polynomial sets.

Findings

Preservers are scalar multiples of affine automorphisms

Results hold for fields of characteristic zero

Extensions to polynomials with roots in number fields or reals

Abstract

Let $K$ be a field of characteristic $0$, and let $k \geq 2$ be an integer. We prove that every $K$-linear bijection $f \colon K[X] \to K[X]$ strongly preserving the set of $k$-free polynomials (or the set of polynomials with a $k$-fold root in $K$) is a constant multiple of a $K$-algebra automorphism of $K[X]$, i.e., that there are elements $a, c \in K^{\times}$ and $b \in K$ such that $f(P)(X) = c P(a X + b)$. When $K$ is a number field or $K=\mathbb R$, we prove that similar statements hold when $f$ preserves the set of polynomials with a root in $K$.