Conjectures inspired in the context of Casas-Alvero conjecture, via the logarithmic derivative of derivatives of a polynomial

Question

In the post (cross-posted in Mathematics Stack Exchange with identificator MSE 4244256 and same title) we assume that $P(x)=a_0+a_{1}x+\ldots+a_{n-1}x^{n-1}+a_{n}x^n$ is a polynomial of degree $1<\deg(P)=d=n$ defined over a field $K$ of characteristic zero. We denote its derivatives as $P^{(i)}(x)$ (writting $P^{(0)}(x)=P(x)$), and $a_n$ denotes the leading coefficient of $P(x)$.

I've stated two conjectures (as speculations from the fact) inspired in that I've proven inductively that each polynomial $p(x)$ of degree $1<\deg(p)$ (and with corresponding $a_n\neq 0$) satisfies $$p(x)=a_n\cdot\left(\frac{n-l}{\frac{d}{dx}\log p^{(l)}(x)} \right)^n\tag{1}$$ for each integer $l$ with $0\leq l<n-1$.

Conjecture 1. Let $P(x)$ a polynomial of degree $1<n$, thus we assume $P^{(n)}(0)\neq 0$. If the equation $$P(x)=\frac{P^{(n)}(0)}{n!}\cdot\left(\frac{n-l}{\frac{d}{dx}\log P^{(l)}(x)} \right)^n$$ holds for each integer $0\leq l<n-1$, then $P(x)$ has the form $$P(x)=\frac{P^{(n)}(0)}{n!}\cdot(x-\alpha)^n$$ for some element $\alpha\in K$.

I know the statement of the called Casas-Alvero conjecture from Wikipedia Casas-Alvero conjecture. I've speculated (while I don't know if this has a good mathematical content, or if these ideas are in the literature in some way more or less explicit) if from previous simple idea one can to state an equivalent form of Casas-Alvero conjecture.

Conjecture 2. Let $P(x)$ a polynomial of degree $1<n$ and leading coefficient $a_n\neq 0$. The Casas-Alvero conjecture is equivalent to that the equation $$P(x)\cdot\left(P^{(l+1)}(x)\right)^n=a_n\left((n-1)P^{(l)}(x)\right)^n,\tag{2}$$ holds for each integer $0\leq l<n-1$.

Question (Updated, considering the kindly advice of moderator in comments). I would like to know what work can be done about the veracity of Conjecture 1, can you prove or refute Conjecture 1? Many thanks.

I don't know if these conjectures Conjecture 1 and Conjecture 2 are in the literature, my idea was very simple: that is to study the logarithmic derivative of derivatives of a given polynomial of the cited form, if it is in the literature please refer it in a comment or in your answer and I try to search and study this from the corresponding articles.

Remark: recently a professor solve a related question [1].

References:

[1] Iterated derivatives and polynomials that are the power of a linear polynomial, Mathematics Stack Exchange (Sep 12, 2021), post with identificator MSE 4248459

Answer 1

I'm probably making a stupid mistake, but is Conjecture 1 possibly rather easy? At $\ell=0$ we're assuming

$$ p(x) = \frac{a_n}{n!} \left( \frac{n}{\frac{d}{dx} \log p(x)} \right)^n $$

Write $\frac{d}{dx} \log p(x) = \dfrac{q(x)}{r(x)}$, a reduced rational expression. Then the right hand side of the hypothesis is

$$ \frac{a_n}{n!} \left( \frac{n r(x)}{q(x)} \right)^n, $$

still a reduced rational expression, and this is equal to $p(x)$, a polynomial. So $q(x)$ is a constant.

Factor $p(x) = \frac{a_n}{n!} (x-r_1)^{e_1} \dotsm (x-r_k)^{e_k}$ (with all $e_i > 0$ and $r_i$ pairwise distinct). Then the logarithmic derivative is

$$ \frac{q(x)}{r(x)} = \frac{d}{dx} \log p(x) = \sum \frac{e_i}{x-r_i} . $$

The numerator of this sum is

$$ \sum_{i=1}^k e_i \prod_{j \neq i} x-r_j . $$

This is a nonzero polynomial (leading coefficient $\sum e_i = n = \deg p$) of degree $k-1$ (the number of distinct roots of $p$) and it's easy to see that its value at each $r_i$ is nonzero, so there will be no cancellation with the denominator. So, this is $q(x)$. It must be that this is constant. So the number of distinct roots of $p$ must be $k=1$, $p$ is a $k$th power.

This was only using the hypothesis at $\ell=0$, not all $0 \leq \ell < n-1$, so perhaps you can tell me if I've made some silly error or misunderstanding.