On principal minors of Bezout matrix

TL;DR

This paper explores the relationship between principal minors of Bezout matrices and divided difference matrices, providing conditions under which the roots of interpolation polynomials are real and separated.

Contribution

It establishes a new relationship between principal minors of Bezout matrices and divided difference matrices, linking matrix properties to root location of interpolation polynomials.

Findings

Principal minors of divided difference matrices indicate real, separated roots.

A relationship between Bezout matrix minors and divided difference minors is established.

Positivity or alternating signs of minors imply real, separated roots of interpolation polynomials.

Abstract

Let $x_1,...,x_{n}$ be real numbers, $P(x)=p_n(x-x_1)...(x-x_n)$, and $Q(x)$ be a polynomial of degree less than or equal to $n$. Denote by $\Delta(Q)$ the matrix of generalized divided differences of $Q(x)$ with nodes $x_1,...,x_n$ and by $B(P,Q)$ the Bezout matrix (Bezoutiant) of $P$ and $Q$. A relationship between the corresponding principal minors, counted from the right-hand lower corner, of the matrices $B(P,Q)$ and $\Delta(Q)$ is established. It implies that if the principal minors of the matrix of divided differences of a function $g(x)$ are positive or have alternating signs then the roots of the Newton's interpolation polynomial of $g$ are real and separated by the nodes of interpolation.