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?


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 $a_i$:
$\mu'_i=\frac 1 n p_i=\frac 1 n \sum_{k=1}^n a_k^i$
$e'_i = \frac {n-i} n e_i$

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 $(n-1)e_1^2 - 2n e_2 = (n-1)(\sum a_j)^2 - 2n \sum_{j<k}a_j a_k$
$= n (\sum a_j)^2 - 2n \sum_{j<k}a_j a_k - (\sum a_j)^2 = n \sum a_j^2 - (\sum a_j)^2$
$= n^2 (\frac 1 n \sum a_j^2 - (\frac {\sum a_j} n)^2) = n^2 \sigma^2$, with $\sigma^2$ the variance of $P$ roots.
So distance between the two roots $= \frac 2 {n!} \sqrt{(n-1)!(n-2)!n^2 \sigma^2} = \frac 2 {\sqrt{n-1}} \sigma$



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