Deterministic Frobenius normal form with transformation in O(n^ω) time

Establish a deterministic algorithm that, given an n×n matrix A over an effective field F, computes the Frobenius normal form of A together with an explicit transformation matrix P such that P^{-1} A P equals the Frobenius form, using O(n^ω) arithmetic operations in F, thereby matching the best-known Las Vegas probabilistic bound.

Background

The paper notes that existing algorithms can compute the Frobenius normal form in O(n^ω) operations using Las Vegas probabilistic methods, but no deterministic algorithm achieving the same bound and producing a transformation matrix is currently known.

The authors present improved deterministic complexity bounds for related tasks but emphasize that achieving O(n^ω) deterministically for both the form and the transformation remains unresolved.