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Sep 21, 2022 at 16:46 vote accept user142929
Sep 21, 2022 at 16:46 comment added user142929 Perfect, many thanks I'm going to accept your answer, and after to study it when I can (I have not internet at home). Many thanks again for these resulsts. I see that your answers are elaborated maybe it is easy to you, but it is worthy.
Sep 21, 2022 at 14:16 comment added Zach Teitler Conjecture 2: the condition (2) , for any single value of $\ell$, implies $P$ is an $n$th power, by unique factorization of polynomials. This is an immediate implication; it is not equivalent to the Casas-Alvero conjecture.
Sep 21, 2022 at 11:47 comment added user142929 Sure that if there is some lack here is from my side. As soon I can I study your excellent answer, many thanks. Feel free to study the veracity of Conjecture 2.
Sep 20, 2022 at 17:25 comment added Zach Teitler (I assumed that $\sum e_i = n = \deg p$ is not divisible by the characteristic of the field.)
Sep 20, 2022 at 17:17 history answered Zach Teitler CC BY-SA 4.0