The Hankel Pencil Conjecture

Abstract

The Toeplitz pencil conjecture stated in [SS1] and [SS2] 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 2 C¤ = Cn f0g; for l ¸ 1: In this paper it is shown to be implied by another conjecture, we call root conjecture. This latter claims for a certain pair (mnn;mn¡1;n) of submaximal minors of certain special Hn(x) that, viewed as elements of C[x]; there holds that roots(mnn) µ roots(mn¡1;n) implies roots(mn¡1;n) = f1g: We give explicit formulae in the ci for these minors and show 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: 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.