Is there a closed-form solution to the following type of system of cubic equations? $$ x_j=\sum_{i=1}^na_{ij}x_i^3,\quad\forall\,j=1,...,n $$ Here the $a_{ij}$'s are constants, and the $x_i$'s are the variables.

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    $\begingroup$ It seems that $n=2$ is awful enough. $\endgroup$ Jul 15, 2022 at 1:25
  • $\begingroup$ $n=2$ actually reduces to a quartic in $x_j^2$ so technically it has closed form solutions, albeit not pretty. No such luck for $n \gt 2$ though, at least not for arbitrary $a_{ij}$. $\endgroup$
    – dxiv
    Jul 15, 2022 at 2:40


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