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?

  • 2
    $\begingroup$ I removed the nt.number-theory tag as your question had nothing to do with number theory. Nice question, by the way! $\endgroup$ – GH from MO Aug 23 '18 at 19:02

By Remark 9.21 page 340 of the book by Basu, Pollack and Roy on real algebraic geometry the matrix $H$ is the expansion of the Bezoutiant of $P$ and $P'$ in the Horner basis of $P$ instead of the basis of usual monomials which is your $M$. The Horner basis is the sequence of polynomials one actually computes when evaluating $P$ by Horner's scheme.


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