I recently came across a criteria to count the number of real zeros of a polynomial $P(x)$ with real coefficients. Unfortunately I cannot find the reference! The criteria is the following: Form the matrix M whose entry $M_{i,j}$ (with $0\leq i,j\leq deg(p)-1$) is the coefficient of $X^iY^j$ in the polynomial $$\frac{P(X)P'(Y)-P(Y)P'(X)}{X-Y}.$$
Then all of the roots of $P$ are real if and only if $M$ is positive semi-definite. Moreover the roots are all distinct if $M$ is positive definite.
This is clearly a variation on Hermite's criteria which says the same thing, but instead of $M$ as above, uses a Hankel matrix $H$ whose entry $H_{i,j}$ (with $i$ and $j$ in the same range) is the symmetric power polynomial $p_{i+j}$ in the roots of the polynomial which can be generated from the coefficients of $P$ using the Newton identities.
The first criteria is much better suited for my specific application and I'd like to cite it, but as I said I can't find the reference. I expect the two matrices are similar, and though it's not immediately obvious to me how, I suspect I can reproduce it with a little effort. The more pressing issue is that I'd like to be able to cite whoever originally discovered this simpler formulation.
Does anyone know a reference for this?
