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

Abstract

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 $g\in G$. For a pseudo-symmetric numerical semigroup, the same result holds if the term $z^{F/2}$ is omitted from $P\left(z\right)$. We give several corollaries, including that the sum of the critical values of a gap polynomial vanishes.