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Suppose $p(x)$ is a univariate real-rooted polynomial. It is easy to see that the following rational polynomial $$\Psi_p(x) = \frac{\partial^2p}{p}(x)=\sum_{1\leq j<k\leq r}\frac{2}{(z_i-\lambda_j)(z_i-\lambda_k)}$$

is nonincreasing when $x>maxroot(p)$, where $\lambda_i,i=1,\cdots,r$ are the roots of $p$.

Now I want a more general case.

Suppose $p(x_1,\cdots,x_n)$ is a multivariate real polynomial and all univariate restrictions are real-rooted. We say $x$ is above the roots of $p$ if $$p(x+t)>0 \text{ for all non-negative vector t}.$$Define

$$\Psi_p^i(x) = \frac{\partial^2_{x_i}p}{p}(x).$$

It is obvious that $\Psi_p^i$ is nonincreasing in $i$ direction when $x$ is above the roots of $p$ for we can consider the univarate restriction of $p$ in that direction and regard it as a univariate polynomial.

What about other direction? Is the rational polynomial above is nonincreasing in any direction when $x$ is above the roots of $p$?

The problem is discribed in a general case. In fact, we can consider the case of only two variables. The problem is about the root of polynomial. Maybe some knowledege of complex analysis is needed.

Another thing I want to mention is that the problem is an variant of the Lemma 17 in this post https://terrytao.wordpress.com/2013/11/04/real-stable-polynomials-and-the-kadison-singer-problem/#more-7109.

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