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 gap polynomial $P\left(z\right)$ of degree $F$ is defined as ${\sum }_{g\in G}{c}_g{z}^g$. We show that for a symmetric numerical semigroup, the power sums of the critical points of its gap polynomials vanish for every power $g\in G$. For pseudo-symmetric numerical semigroups, the same result holds if the term $z^{F\mathrm{/}2}$ is also omitted from $P\left(z\right)$. We give several corollaries, including the fact that the sum of the critical values of a gap polynomial vanishes.