  We revisit the notion of root polynomials, thoroughly studied in [F. Dopico
and V. Noferini, Root polynomials and their role in the theory of matrix
polynomials, Linear Algebra Appl. 584:37--78, 2020] for general polynomial
matrices, and show how they can efficiently be computed in the case of matrix
pencils. The staircase algorithm implicitly computes so-called zero directions,
as defined in [P. Van Dooren, Computation of zero directions of transfer
functions, Proceedings IEEE 32nd CDC, 3132--3137, 1993]. However, zero
directions generally do not provide the correct information on partial
multiplicities and minimal indices. These indices are instead provided by two
special cases of zero directions, namely, root polynomials and vectors of a
minimal basis of the pencil. We show how to extract, starting from the block
triangular pencil that the staircase algorithm computes, both a minimal basis
and a maximal set of root polynomials in an efficient manner. Moreover, we
argue that the accuracy of the computation of the root polynomials can be
improved by making use of iterative refinement.
