Krylov-like method for solving systems of polynomials?

Sort of, the root finding problem is equivalent to the eigenvalue problem associated with the companion matrix. Nonsymmetric eigenvalue methods such as "Krylov-Schur" can be used here.

Notes:

The monic polynomials are extremely ill-conditioned and thus a better conditioned polynomial basis is mandatory for moderate to high order.
The companion matrix is already Hessenberg.

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