Tell me more ×  Hi there,  I am trying to find complex solutions
with positive real part $\{t_j \;|\;{\rm Re}\;t_j>0, j = 1, 2,
3, \dots, n\}$ of the system of equations $$0 = 1 + \sum_j
\left(t_j^{2l+1} + {t_j^*}^{2l+1}\right),\; l = 1,2,3,\dots m.$$
Where for a given $n$ I would like to make $m$ as large as
possible. Since, this system is non-analytic and thus for $n=m$
most likely under-constrained, I had the idea to just fix the
magnitude of all solutions to 1: $t_j = e^{i\phi_j}$ with
$-{\pi\over 2} < \phi_1 \le \phi_2 \le \dots \le \phi_n
<{\pi\over 2}$. In terms of these the system becomes: $$0 = 1 +
2\sum_j \cos{\left[\phi_j(2l+1)\right]},\; l = 1,2,3,\dots n.$$
This definitely has solutions up to $n=m=2$, but already for
$n=3$, my naive attempt at numerically solving this
(Mathematica's NSolve) is taking quite long. Is there some
better way to find or at least confirm the existence of such
solutions?  Thanks, Nik  share|improve this question What is
$t_j^*$? Is that the complex conjugate? –  Gerry My