The Hankel pencil conjecture
For \(n\geq3\) consider the Hankel \(n\times n\) matrix \(H_{n}(x)=H_{n} (x;c_{1},\dots ,c_{n+1})\) over \(C[x]\) where the \((i,j)\)th entry is equal to \(x\) if \(i+j=n-1\); to \(c_{i+j-n+1}\) if \(i+j\geq n\); and to \(0\) otherwise. The ``Hankel pencil conjecture'' (HPnC) states that if \(H_{n}(x)=0\) and \(c_{1},\dots ,c_{n+1}\) are nonzero complex numbers, then the last two columns of \(H_{n}(x)\) are linearly dependent. This conjecture was proved for \(n\leq5\) by \textit{W. Schmale} and \textit{P. K. Scharma} [Linear Algebra Appl. 389, 33--42 (2004; Zbl 1068.15008)], but a purported proof by V. Bolotnikov and C. K. Li of the general case has been shown to have a significant gap. The authors of the present paper show that for degree \(n\) HPnC is implied by a new conjecture RnC (the ``root conjecture'') concerning the roots of certain polynomials of degree \(n-2\). Computations are used to verify RnC (and hence HPnC) for all \(n\leq8\).
scientific article; zbMATH DE number 5603569