Power sums of critical points of gap polynomials for symmetric and pseudo-symmetric numerical semigroups

For a numerical semigroup $S$ with Frobenius number $F$ and gaps $G$, a polynomial $P\left(z\right)=\sum_{g\in G}c_{g}z^{g}$ of degree $F$ is called a gap polynomial. We show that for a symmetric numerical semigroup, the power sums of the critical points of a gap polynomial vanish for every power $g