Injectivity of the matrix evaluation map in the borderline case m=1 and n<d

Determine whether, for a field F and a polynomial f in F[x] with degree greater than 1, the evaluation map eva_{f, M_n(F)}: M_n(F) -> M_n(F), A -> f(A), is injective under the following conditions: write g(x) = f(x) - f(0), factor g(x) = x^m h(x) with m >= 1 maximal and h(0) != 0, and let d be the minimal degree among the irreducible factors of h(x) in F[x]; establish injectivity in the case m = 1 and n < d.

Background

Theorem ‘matrix’ analyzes when the evaluation map A -> f(A) on M_n(F) can be injective for a non-linear polynomial f. Writing g(x) = f(x) - f(0) and factoring g(x) = x^m h(x) with m >= 1 maximal and h(0) != 0, the theorem proves non-injectivity when m >= 2, and also when m = 1 and n >= d, where d is the minimal degree among the irreducible factors of h(x).

In the remaining borderline case m = 1 and n < d, the theorem shows only that no nonzero matrix A satisfies f(A) = f(0) I_n, but it does not settle whether eva_{f, M_n(F)} is injective. The posed problem seeks to determine injectivity precisely in this unresolved regime, which would complete the classification initiated by the theorem.