Reconstructing Numerical Semigroups from Circulant Spectral Dimensions
Abstract
The polynomial algebra generated by the spectral moments of a family of circulant graphs consists of all eventually polynomial sequences in the graph order; as an unfiltered algebra, it therefore carries little information about the jumps. We show that its natural power-sum weight filtration retains substantially finer arithmetic data. For a finite jump set S with gcd(S)=1 and M=max S, the relevant invariant is the numerical semigroup ΣS=⟨ M,M−s,M+s:s∈ S⟩, which arises from the lengths of closed walks winding around the cycle. We prove that the eventual global filtered dimension is MD+⌊ D/2⌋+1−M(M+1)−g(ΣS). Restricting graph orders to n≡ apmod q, for q| M, replaces the genus term by the number of gaps of ΣS in the class −apmod q; the class a=0 gives the genus of the quotient semigroup ΣS/q. These formulas yield an inverse reconstruction theorem: the unrestricted dimensions determine M, and the dimensions in all classes modulo M at one common stable degree determine the Apéry set and hence ΣS. We give effective thresholds, exact examples, and a finite-rank algorithm valid in every degree. Reconstruction of the semigroup need not identify the original jump set.
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BibTeX
@misc{caicedo2026reconstructing,
title = {{Reconstructing Numerical Semigroups from Circulant Spectral Dimensions}},
author = {Juan Diego Velez Caicedo and Carlos A. Cadavid},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.14316},
eprint = {2611.14316},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.14316},
}
RIS
TY - UNPB
TI - Reconstructing Numerical Semigroups from Circulant Spectral Dimensions
AU - Juan Diego Velez Caicedo
AU - Carlos A. Cadavid
PY - 2026
DA - 2026/11/10
PB - arXiv
DO - 10.48550/arxiv.2611.14316
UR - https://arxiv.org/abs/2611.14316
AB - The polynomial algebra generated by the spectral moments of a family of circulant graphs consists of all eventually polynomial sequences in the graph order; as an unfiltered algebra, it therefore carries little information about the jumps. We show that its natural power-sum weight filtration retains substantially finer arithmetic data. For a finite jump set S with gcd(S)=1 and M=max S, the relevant invariant is the numerical semigroup Σ_S=⟨ M,M−s,M+s:s∈ S⟩, which arises from the lengths of closed walks winding around the cycle. We prove that the eventual global filtered dimension is MD+⌊ D/2⌋+1−M(M+1)−g(Σ_S). Restricting graph orders to n≡ apmod q, for q| M, replaces the genus term by the number of gaps of Σ_S in the class −apmod q; the class a=0 gives the genus of the quotient semigroup Σ_S/q. These formulas yield an inverse reconstruction theorem: the unrestricted dimensions determine M, and the dimensions in all classes modulo M at one common stable degree determine the Apéry set and hence Σ_S. We give effective thresholds, exact examples, and a finite-rank algorithm valid in every degree. Reconstruction of the semigroup need not identify the original jump set.
ER -