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

  • Pavel Grinfeld
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Abstract

For a numerical semigroup S with Frobenius number F and gaps G, a polynomial P(z)=∑g∈ Gcgzg 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∈ G. For a pseudo-symmetric numerical semigroup, the same result holds if the term zF/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.

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Cite

BibTeXRIS
BibTeX
@misc{grinfeld2026power,
  title = {{Power sums of critical points of gap polynomials for symmetric and pseudo-symmetric numerical semigroups}},
  author = {Pavel Grinfeld},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.03049},
  eprint = {2609.03049},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.03049},
  note = {CC0},
}
RIS
TY  - UNPB
TI  - Power sums of critical points of gap polynomials for symmetric and pseudo-symmetric numerical semigroups
AU  - Pavel Grinfeld
PY  - 2026
DA  - 2026/09/04
PB  - arXiv
DO  - 10.48550/arxiv.2609.03049
UR  - https://arxiv.org/abs/2609.03049
AB  - For a numerical semigroup S with Frobenius number F and gaps G, a polynomial P(z)=∑_g∈ Gc_gz^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∈ 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.
ER  -