Characterization of a family of generalized companion matrices

Matrices A of order n having entries in the field F(x1,…,xn) of rational functions over a field F and characteristic polynomialdet⁡(tI−A)=tn+x1tn−1+⋯+xn−1t+xn are studied. It is known that such matrices are irreducible and have at least 2n−1 nonzero entries. Such matrices with exactly 2n−1 nonzero entries are called Ma-Zhan matrices. Conditions are given that imply that a Ma-Zhan matrix is similar via a monomial matrix to a generalized companion matrix (that is, a lower Hessenberg matrix with ones on its superdiagonal, and exactly one nonzero entry in each of its subdiagonals). Via the Ax-Grothendieck Theorem (respectively, its analog for the reals) these conditions are shown to hold for a family of matrices whose entries are complex (respectively, real) polynomials.