# Quantify the similarity between a polynomial roots and the roots of its derivatives

On $$\mathbb{C}[X]$$, many theorems and conjectures deal with relations between a polynomial roots and the roots of its derivatives. When looking at a graph, the derivative roots distribution somewhat mimics the distribution of the polynomial roots. It is this "somewhat mimics" that I would like to look at in this question.

Examples. Let $$P$$ be a degree $$n$$ polynomial in $$\mathbb{C}[X]$$, then:

• (Well-known) The mean of $$P$$ roots is also the mean of $$P$$ successive derivatives roots. And so it is the only root of $$P^{(n-1)}$$ which has degree $$1$$.
• The mean being the first cumulant, what about the other cumulants of the roots? The second cumulant is the variance $$\sigma^2$$, same as second central moment. We find that $$\sigma^2/(n-1)$$ is conserved: if $$\sigma'^2$$ is the variance of $$P'$$ roots, $$\sigma'^2 = \frac {n-2} {n-1} \sigma^2$$ (proof at the end).
• Similarly, for the third cumulant $$\kappa_3$$, which is also the third central moment $$\mu_3$$: if $$\kappa_3'$$ is the third cumulant of $$P'$$ roots, $$\kappa_3'= \frac {n-3} {n-1} \kappa_3$$ (proof at the end). However this does not extend to $$\kappa_4$$ nor to $$\mu_4$$ (which by the way are different).
• $$\sigma'^2 = \frac {n-2} {n-1} \sigma^2$$ has the following consequence (proof at the end): distance between the two roots of $$P^{(n-2)}$$ is proportional to $$\sigma$$: $$\frac 2 {\sqrt{n-1}} \sigma$$.

Question: Are there other quantities that characterize the roots distribution of a polynomial, that are conserved (possibly with a factor only depending upon $$n$$) in the polynomial derivative?
These quantities should have an established statistical meaning, or a geometric interpretation.
E.g. what about the PCA (principal component analysis) of the roots?

Proofs:

Use the following relations between cumulants $$\kappa_i$$, elementary symmetrical polynomials $$e_i$$, elementary symmetric polynomials for the derivative $$e'_i$$, central moments $$\mu_i$$, raw moments $$\mu'_i$$ (sorry for the notation clash: not the central moments of the derivative), power sums $$p_i$$, polynomial roots $$a_i$$:
$$\mu'_i=\frac 1 n p_i=\frac 1 n \sum_{k=1}^n a_k^i$$
$$p_1=e_1$$
$$p_2=e_1^2-2e_2$$
$$p_3=e_1^3-3e_1e_2+3e_3$$
$$\kappa_2=\mu'_2-\mu_1'^2$$
$$\kappa_3=\mu'_3-3\mu'_2\mu'_1+2\mu_1'^3$$
$$e'_i = \frac {n-i} n e_i$$

Second cumulant (variance):
$$\kappa_2=\mu'_2-\mu_1'^2=\frac 1 n p_2 - \frac 1 {n^2} p_1^2$$
$$=\frac {n-1} {n^2} e_1^2 - \frac 2 n e_2$$
$$\kappa'_2=\frac {n-2} {(n-1)^2} e_1'^2 - \frac 2 {n-1} e'_2$$
then replace $$e'_i$$ with $$\frac {n-i} n e_i$$, gives $$\kappa'_2=\frac {n-2} {n-1} \kappa_2$$.

Third cumulant: $$\kappa_3=\mu'_3-3\mu'_2\mu'_1+2\mu_1'^3$$
$$=\frac 1 n p_3 - \frac 3 {n^2} p_2p_1 + \frac 2 {n^3} p_1^3$$
$$= \frac 1 n (e_1^3-3e_1e_2+3e_3) - \frac 3 {n^2}(e_1^2-2e_2)e_1 + \frac 2 {n^3}e_1^3$$
$$=\frac {(n-1)(n-2)} {n^3} e_1^3 - 3 \frac {n-2} {n^2}e_1e_2 + \frac 3 n e_3$$
$$\kappa'_3= \frac {(n-2)(n-3)} {(n-1)^3} e_1'^3 - 3 \frac {n-3} {(n-1)^2}e'_1e'_2 + \frac 3 {n-1} e'_3$$
then replace $$e'_i$$ with $$\frac {n-i} n e_i$$, gives $$\kappa'_3=\frac {n-3} {n-1} \kappa_3$$.

Distance between roots of the $$(n-2)$$th derivative: this can be proven using the variance conservation relation, or directly:
Let $$P(Z) = \sum_{j=0}^n (-1)^j e_j \; Z^{n-j}$$.
$$P^{(n-2)}(Z)=\frac {n!} 2 Z^2-(n-1)! \; e_1 \; Z+(n-2)! \; e_2.$$
Its two roots are $$\frac 1 n {e_1} \pm \frac 1 {n!} \sqrt{(n-1)!(n-2)!((n-1)e_1^2-2ne_2)}$$.
The quantity $$(n-1)e_1^2 - 2n e_2 = (n-1)(\sum a_j)^2 - 2n \sum_{j
$$= n (\sum a_j)^2 - 2n \sum_{j
$$= n^2 (\frac 1 n \sum a_j^2 - (\frac {\sum a_j} n)^2) = n^2 \sigma^2$$, with $$\sigma^2$$ the variance of $$P$$ roots.
So distance between the two roots $$= \frac 2 {n!} \sqrt{(n-1)!(n-2)!n^2 \sigma^2} = \frac 2 {\sqrt{n-1}} \sigma$$