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Given positive integer $n$. Let perfect polynomials be polynomials on $\mathbb{C}$ which can be written as $P_1^n+\cdots+P_n^n$, where $P_i\in\mathbb{C}[x]$. Prove that the product of $n$ perfect polynomials is still perfect number.

I think the proof is to directly construct $P_i$ of the product. For example, when $n=2$, we have $(P_1^2+P_2^2)(Q_1^2+Q_2^2)={(P_1Q_1+P_2Q_2)}^2+{(P_1Q_2-P_2Q_1)}^2$. I want to find out if there are similiar equations in the case of $n\ge3$.

It is similiar to my last question, which was proved false. However, I still want to discuss on $\mathbb{C}$.

P.S. Actually, every complex polynomial is perfect, but please don't prove the statement in this way.

OrthoPole
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  • The link to my last question: https://math.stackexchange.com/questions/4317390/an-elementary-identical-equation – OrthoPole Nov 29 '21 at 11:14
  • It would be more natural to continue your previous question for polynomials with integer coefficients. To "forbid" us to use the obvious conclusion that every complex polynomial is "perfect" (not a good term for integers, by the way), doesn't seem reasonable. – Dietrich Burde Nov 29 '21 at 12:09
  • Firstly, the statement is false when it is limited to integer coefficients, with reference to the answer to my last question. Also, it is not obvious that every complex polynomial is perfect (maybe you can have a try). My purpose is to find an equation that fits the requirements, instead of solving the problem. – OrthoPole Nov 29 '21 at 12:20

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