Let $p(x)$ be a polynomial of degree $n>2$, with roots $x\_1,x\_2,\dots,x\_n$ (including multiplicities). Let $m$ be a positive even integer. Define the following mapping
$$V\_m(p)=\sum\_{1\leq i<j\leq n}(x\_i-x\_j)^m.$$



> 
> **QUESTION.** For $\deg p(x)=n>2$ and $p'(x)$ its derivative, can you express
> $$\frac{V\_m(p)}{V\_m(p')}$$
> as a function of $m$ and $n$ alone?
> 
> 
> 


**Remark.** Prompted by Fedor's questions, as a showcase I just computed (not proved) that
$$\frac{V\_2(p)}{V\_2(p')}=\frac{n^2}{(n-1)(n-2)}.$$



Ratios of polynomials and derivatives under a certain functional
Here a [SageMath code](https://sagecell.sagemath.org/?z=eJx9jrFuwkAQRPv7ipFSsBsMxJRILnGHlEskNxGJHDis03n3osMu_Pc54qB02W6kN2_WWkWFOrWnwUetvevP9Bz7SaP4tn_x2pG1BRa6YDY31Fpdd06JzSGn10nEDcmf6lF_DNeMK68lzoKM7f_BXO_E6dCmKYPGnN0FDQnv8HcPaD6EvtjcQsqu6ygEWpX8HvCIT_87JEXgnA9vsgpFOOISEwK8IrXaOSoLyGazxRIl4y5bViDNprl3vA-G6mkG3DAmxZ5Sfq6hLX8DGQJUpQ==&lang=sage&interacts=eJyLjgUAARUAuQ==) that provides a function `V(m)` computing $V\_m(p)$ in terms of elementary symmetric functions of $x\_1,\dots,x\_n$ (i.e. coefficients of $p$).


For example, if $p(x) = x^n - e\_1 x^{n-1} + e\_2 x^{n-2} + \dots$, then
$$V\_2(p) = (n-1)e\_1^2 - 2n e\_2,$$
$$V\_4(p) = (n-1) e\_1^4 - 4n e\_2 e\_1^2 + (2n+12)e\_2^2 + (4n-12) e\_3 e\_1 - 4n e\_4,$$
and so on.


From these expressions a proof for $m=2$ follows instantly. However, for larger $m$ the ratio $\frac{V\_m(p)}{V\_m(p')}$ does not appear to be a function of $n$, which I've tested computationally for $m$ up to $20$.


