Let $m_\lambda(X_1(t),X_2(t),...X_N(t))$ be a monomial symmetric function with partition $\lambda$.
For example:
$ m_{(3,1,1)}(X_1(t),X_2(t),...X_N(t)) =X_1^3X_2X_3 + X_1X_2^3X_3 + X_1X_2X_3^3 $
If $X_j=e^{i \omega_j t}$, is it possible to prove that there is at least one root $m_\lambda(t)=0$?
If not, is it possible with some restrictions on the set $\{\omega_j\}$?