Power sums of critical points of gap polynomials for symmetric and pseudo-symmetric numerical semigroups
Year: 2026
Type: preprint
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.
Source: arXiv (Cornell University)
Author: Grinfeld, Pavel
Institution: Drexel University
Language: English
Subfield: Algebra and Number Theory
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7208726083