MATRIX CYCLIZATION OVER COMPLEX POLYNOMIALS

Let A be an n x n, B an n x m complex polynomial matrix. The following open problem is investigated: does there exist a complex polynomial ( m x n)-matrix F such that A + BF has a cyclic vector in the image of B ? An explicit solution of the problem is given for the following gener ic situation: n greater than or equal to 4, [B,AB,...,A(n-1)B] is righ tinvertible (necessary), the entries of a specific normalized form of (A,B) do not satisfy certain polynomial equations, and one specific en try has at least one simple zero outside a certain finite set. The exc eptional equations are given explicitely and can easily be checked for . (C) 1998 Elsevier Science Inc. All rights reserved.