On principal minors of Bezout matrix
This provides a new theoretical connection in polynomial theory, relevant for researchers in numerical analysis and linear algebra.
The paper establishes a relationship between principal minors of the Bezout matrix and the matrix of generalized divided differences, showing that positivity or alternating signs of these minors imply that the roots of Newton's interpolation polynomial are real and separated by the interpolation nodes.
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 $Δ(Q)$ 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 $Δ(Q)$ 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.