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Existence of nontrivial root-preserving linear bijections over non-algebraically-closed fields

Determine whether there exists a field K that is not algebraically closed and a K-linear bijection f: K[X] → K[X] that preserves the set of polynomials having at least one root in K but is not of the form f(P)(X) = c · P(aX + b), equivalently, f ∉ K× ⋊ Aut_{K-alg}(K[X]); in particular, investigate the case where K is a non-Archimedean local field.

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Background

The paper establishes that for number fields and for K = R, every K-linear bijection on K[X] that preserves the set of polynomials with a root in K must be an affine change of variables up to a nonzero scalar (i.e., lies in K× ⋊ Aut_{K-alg}(K[X])).

It remains unclear whether there are other (non–algebraically closed) fields admitting root-preserving linear bijections that are not of this standard form. The authors specifically suggest beginning with non-Archimedean local fields as a test case.

References

Other natural questions which are left for further study are the following: Is there a field $K$ which is not algebraically closed, and such that there is a $K$-linear bijection $K[X] \to K[X]$, preserving the set of polynomials with a root in~$K$, which is not in $K{\times} \rtimes Aut_{K -alg} (K[X])$? Settling the case of non-Archimedean local fields would be a good start.

— Symmetries of various sets of polynomials (2407.09118 - Seguin, 12 Jul 2024) in Section 1.2 (Main results), end; under “Other natural questions which are left for further study”