The Hankel pencil conjecture

The Toeplitz pencil conjecture stated in [W. Schmale, P.K. Sharma, Problem 30-3: singularity of a toeplitz matrix, IMAGE 30 (2003); W. Schmale, P.K. Sharma, Cyclizable matrix pairs over C[x] and a conjecture on toeplitz pencils, Linear Algebra Appl. 389 (2004) 33-42] is equivalent to a conjecture for n×n Hankel pencils of the form Hn(x)=(ci+j-n+1), where c0=x is an indeterminate, cl=0 for l<0, and cl∈C∗=C⧹{0}, for l⩾1. In this paper it is shown to be implied by another conjecture, which we call the root conjecture. The root conjecture asserts a strong relationship between the roots of certain submaximal minors of Hn(x) specialized to have c1=c2=1. We give explicit formulae in the ci for these minors and prove the root conjecture for minors mnn,mn-1,n of degree ⩽6. This implies the Hankel Pencil conjecture for matrices up to size 8×8. The main tools involved are a partial parametrization of the set of solutions of systems of polynomial equations that are both homogeneous and index sum homogeneous, and use of the Sylvester identity for matrices.