To iteratively solve large linear systems, many current state-of-the-art methods work by finding approximate solutions in successively larger (Krylov) subspaces. Are there similar iterative methods for solving systems of polynomial equations by finding approximate solutions on successively larger algebraic sets?
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.
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.