It is well know that some theorems for polynomials have analogous for integers.
Example. The Mason–Stothers theorem, see this Wikipedia and the abc conjecture for integers.
After I've read the statement for fields of characteristic zero of Casas-Alvero conjecture I wondered next questions.
Question 1. Can you provide us some idea why is it difficult to prove or refute such the Casas-Alvero conjecture? I am asking from an informative viewpoint, that if you can provide me some idea why such conjecture is so difficult. I wish you a good day.
For this next question, I don't know if it have full mathematical sense. If this is the issue please explain why the analogous Casas-Alvero conjecture for positive integers have no mathematical meaning.
Question 2. Has mathematical meaning and what should be an analogous statement , than Casas-Alvero conjecture, for integers? Many thanks.
Question 1: There is a good survey in the article Constraints on counterexamples to the Casas-Alvero conjecture, and a verification in degree 12 , which gives some idea why it is difficult to prove or refute this conjecture. Phrases like "Because of a lack of a general strategy" support this, too. The conjecture has been proven for prime power degree. The smallest unknown cases are degrees $12, 20, 24$ and $28$.
Question 2: An analogue for integers could start from the arithmetic derivative, which is a version of "derivative" for integers. I have not seen a "reasonable" version of Casas-Alvero for integers so far, in contrast to Fermat for polynomials, Mason-Stothers and Waring's problem.