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I have been facing a problem and I cannot understand if this is trivial or not.

Consider the following equation :

$$\sum_{\sigma\in\mathfrak{S}_3}\epsilon(\sigma)n_{\sigma(1)}^2n_{\sigma(2)}=0 $$

$$\text{ or equivalently : }n_1^2n_2+n_2^2n_3+n_3^2n_1=n_2^2n_1+n_3^2n_2+n_1^2n_3 $$

where $n_1,n_2,n_3$ are three non-zero positive integers. Clearly the set $\{n_1=n_2\}\cup\{n_1=n_3\}\cup\{n_2=n_3\}$ gives us trivial solutions.

I would like to know if there exists a non-trivial solution to this equation, any help will be appreciated. Of course one could also consider more generally, given a homogenous polynomial $P$ with $k$ variables something like :

$$\sum_{\sigma\in\mathfrak{S}_k}\epsilon(\sigma)\sigma.P(n_1,...,n_k)=0 $$

But this is (I think) hopeless to understand this generally.

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Your equation factorises as $(n_1-n_2)(n_2-n_3)(n_3-n_1) = 0$

So there are no non-trivial solutions.

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  • $\begingroup$ This also answers the first question, it is (almost) trivial. Thanks. $\endgroup$ – Clément Guérin Sep 18 '15 at 15:24

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