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



Solving a system of algebraic equations
I would try to get rid of the trigonometric functions, and rather rewrite the system as a polynomial system. If $x\_j=\cos(\phi\_j)$, then $\cos(\phi\_j(2\ell+1))=T\_{2\ell+1}(x\_j)$, where $T\_k$ is the $k$-th Chebyshev polynomial of degree $k$. So your system of equations is
\begin{equation}
0=1+2\sum\_j T\_{2\ell+1}(x\_j),\;\;l=1,2,\dots,m,
\end{equation}
with the requirement that $0<x\_j\le 1$. Such a system can be discussed via Groebner bases, see [here](http://ask.sagemath.org/question/403/method-for-solving-a-large-system-of-under) how to do that with Sage. For $m=n\le4$ there are only finitely many solutions. Among them pick those which fit your inequalities. For instance, for $m=n=4$, an approximation of a solution seems to be


$$\begin{align\*}
x\_1 &= 0.963494595276259\\
x\_2 &= 0.852773246361416\\
x\_3 &= 0.600336170417163\\
x\_4 &= 0.058262327046178
\end{align\*}
$$


Of course, you can use this technique also to handle the original case, where $t\_j$ need not have length $1$. For instance, for $n=2$ one can show that $m\le4$, and and approximate solution for $m=4$ is $t\_1=0.466916296430820 + 0.717248344919154i$, $t\_2=0.856453001234213 + 0.445264622297009i$.


One cannot expect simple expressions for the $t\_j$. For instance, the absolute values of the $t\_j$ are roots of an irreducible (over the rationals) polynomial of degree $60$.


