Fel's Conjecture on Syzygies of Numerical Semigroups
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arXiv
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2026
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| author | Chen, Evan Cummins, Chris GSM Grubisic, Dejan Haller, Leopold Hong, Letong Kurghinyan, Andranik Lau, Kenny Leather, Hugh Lee, Seewoo Markosyan, Aram Ono, Ken Patel, Manooshree Pendharkar, Gaurang Rathi, Vedant Schneidman, Alex Seeker, Volker Sengupta, Shubho Sinha, Ishan Xin, Jimmy Zhang, Jujian |
| author_facet | Chen, Evan Cummins, Chris GSM Grubisic, Dejan Haller, Leopold Hong, Letong Kurghinyan, Andranik Lau, Kenny Leather, Hugh Lee, Seewoo Markosyan, Aram Ono, Ken Patel, Manooshree Pendharkar, Gaurang Rathi, Vedant Schneidman, Alex Seeker, Volker Sengupta, Shubho Sinha, Ishan Xin, Jimmy Zhang, Jujian |
| contents | Let $S=\langle d_1,\dots,d_m\rangle$ be a numerical semigroup and $k[S]$ its semigroup ring. The Hilbert numerator of $k[S]$ determines normalized alternating syzygy power sums $K_p(S)$ encoding alternating power sums of syzygy degrees. Fel conjectured an explicit formula for $K_p(S)$, for all $p\ge 0$, in terms of the gap power sums $G_r(S)=\sum_{g\notin S} g^r$ and universal symmetric polynomials $T_n$ evaluated at the generator power sums $σ_k=\sum_i d_i^k$ (and $δ_k=(σ_k-1)/2^k$). We prove Fel's conjecture via exponential generating functions and coefficient extraction, solating the universal identities for $T_n$ needed for the derivation. The argument is fully formalized in Lean/Mathlib, and was produced automatically by AxiomProver from a natural-language statement of the conjecture. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2602_03716 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Fel's Conjecture on Syzygies of Numerical Semigroups Chen, Evan Cummins, Chris GSM Grubisic, Dejan Haller, Leopold Hong, Letong Kurghinyan, Andranik Lau, Kenny Leather, Hugh Lee, Seewoo Markosyan, Aram Ono, Ken Patel, Manooshree Pendharkar, Gaurang Rathi, Vedant Schneidman, Alex Seeker, Volker Sengupta, Shubho Sinha, Ishan Xin, Jimmy Zhang, Jujian Combinatorics Commutative Algebra Number Theory 20M14, 11P91 Let $S=\langle d_1,\dots,d_m\rangle$ be a numerical semigroup and $k[S]$ its semigroup ring. The Hilbert numerator of $k[S]$ determines normalized alternating syzygy power sums $K_p(S)$ encoding alternating power sums of syzygy degrees. Fel conjectured an explicit formula for $K_p(S)$, for all $p\ge 0$, in terms of the gap power sums $G_r(S)=\sum_{g\notin S} g^r$ and universal symmetric polynomials $T_n$ evaluated at the generator power sums $σ_k=\sum_i d_i^k$ (and $δ_k=(σ_k-1)/2^k$). We prove Fel's conjecture via exponential generating functions and coefficient extraction, solating the universal identities for $T_n$ needed for the derivation. The argument is fully formalized in Lean/Mathlib, and was produced automatically by AxiomProver from a natural-language statement of the conjecture. |
| title | Fel's Conjecture on Syzygies of Numerical Semigroups |
| topic | Combinatorics Commutative Algebra Number Theory 20M14, 11P91 |
| url | https://arxiv.org/abs/2602.03716 |