Solvability of the Hankel determinant problem for real sequences

TL;DR

This paper investigates the conditions under which Hankel determinants derived from real sequences are solvable, providing new proofs and insights into the structure of Hankel matrices and their associated polynomials.

Contribution

It offers a simplified proof of Kronecker's classical result and characterizes sequences that generate Hankel determinants, advancing understanding of the Hankel determinant problem.

Findings

Short proof of Kronecker's result on Hankel matrix rank

Characterization of sequences with given Hankel determinants

New proof of a theorem on positive semidefiniteness of Hankel matrices

Abstract

To each nonzero sequence $s:= \{s_{n}\}_{n \geq 0}$ of real numbers we associate the Hankel determinants $D_{n} = \det \mathcal{H}_{n}$ of the Hankel matrices $\mathcal{H}_{n}:= (s_{i + j})_{i, j = 0}^{n}$, $n \geq 0$, and the nonempty set $N_{s}:= \{n \geq 1 \, | \, D_{n-1} \neq 0 \}$. We also define the Hankel determinant polynomials $P_0:=1$, and $P_n$, $n\geq 1$ as the determinant of the Hankel matrix $\mathcal H_n$ modified by replacing the last row by the monomials $1, x, \ldots, x^n$. Clearly $P_n$ is a polynomial of degree at most $n$ and of degree $n$ if and only if $n\in N_s $. Kronecker established in 1881 that if $N_s $ is finite then rank $\mathcal{H}_{n} = r$ for each $n \geq r-1$, where $r := \max N_s $. By using an approach suggested by I.S.Iohvidov in 1969 we give a short proof of this result and a transparent proof of the conditions on a real sequence $\{t_n\}_{n\geq 0}$ to be of the form $t_n=D_n$, $n\geq 0$ for a real sequence $\{s_n\}_{n\geq 0}$. This is the Hankel determinant problem. We derive from the Kronecker identities that each Hankel determinant polynomial $ P_n $ satisfying deg$P_n = n\geq 1$ is preceded by a nonzero polynomial $P_{n-1}$ whose degree can be strictly less than $n-1$ and which has no common zeros with $ P_n $. As an application of our results we obtain a new proof of a recent theorem by Berg and Szwarc about positive semidefiniteness of all Hankel matrices provided that $D_0 > 0, \ldots, D_{r-1} > 0 $ and $D_n=0$ for all $n\geq r$.