Positive-characteristic extension of the k-free/k-fold-root linear preserver classification

Determine whether, for fields K of positive characteristic and any integer k ≥ 2, every K-linear bijection f: K[X] → K[X] that preserves either (i) the set of k-free polynomials in K[X] or (ii) the set of polynomials having a k-fold root in K must necessarily be of the form f(P)(X) = c · P(aX + b) for some a, b in K and c in K×, as in the characteristic-zero case.

Background

The paper proves that in characteristic zero, any K-linear bijection f: K[X] → K[X] preserving either the set of k-free polynomials or the set of polynomials with a k-fold root in K must be an affine change of variables composed with a constant multiple, i.e., f(P)(X) = c * P(aX + b).

This result relies on techniques that use characteristic-zero tools (e.g., repeated differentiation and arguments that fail when characteristic divides certain integers). The authors note that a key lemma used in their approach fails in positive characteristic when p | k, leaving open whether an analogous classification still holds more generally in positive characteristic.