Let S be the numerical semigroup under consideration, with multiplicity m, generators d_1,…,d_m, gap power sums G_r(S), and invariants K_p(S). Write sigma=(sigma_1,sigma_2,…) and delta=(delta_1,delta_2,…), where sigma_k=∑_i=1^m d_i^k,qquad delta_k=fracsigma_k-12^k. Let T_n be the universal symmetric polynomials defined from these power sums. Fel's conjecture. For every p≥ 0, K_p(S)=∑_r=0^p binomprT_p-r(sigma)G_r(S)+frac2^p+1p+1T_p+1(delta). This conjecture gives a complete formula for the invariants K_p(S) in terms of the universal symmetric polynomials and the gap power sums; the displayed cases for p≤ 3 provide the initial evidence, while the general identity is the conjectural part.

Fel's conjecture on universal symmetric polynomials for numerical semigroups

About 1 year old · traced to 2025

Let $S$ be the numerical semigroup under consideration, with multiplicity $m$ , generators $d_1,\dots,d_m$ , gap power sums $G_r(S)$ , and invariants $K_p(S)$ . Write $\sigma=(\sigma_1,\sigma_2,\dots)$ and $\delta=(\delta_1,\delta_2,\dots)$ , where

\sigma_k=\sum_{i=1}^m d_i^k,\qquad \delta_k=\frac{\sigma_k-1}{2^k}.

Let $T_n$ be the universal symmetric polynomials defined from these power sums. Fel's conjecture. For every $p\geq 0$ ,

K_p(S)=\sum_{r=0}^p\binom{p}{r}T_{p-r}(\sigma)G_r(S)+\frac{2^{p+1}}{p+1}T_{p+1}(\delta).

This conjecture gives a complete formula for the invariants $K_p(S)$ in terms of the universal symmetric polynomials and the gap power sums; the displayed cases for $p\leq 3$ provide the initial evidence, while the general identity is the conjectural part.

References

Primary source

Evan Chen, Chris Cummins, GSM, Dejan Grubisic, Leopold Haller, Letong Hong, Andranik Kurghinyan, Kenny Lau, Hugh Leather, Seewoo Lee, Aram Markosyan, Ken Ono, Manooshree Patel, Gaurang Pendharkar, Vedant Rathi, Alex Schneidman, Volker Seeker, Shubho Sengupta, Ishan Sinha, Jimmy Xin and Jujian Zhang, “Fel's Conjecture on Syzygies of Numerical Semigroups”, arXiv:2602.03716 (2026).

Additional references

2 papers in this index state this conjecture (2025–2026). The statement above is taken from the most recent of them; the others are arXiv:2507.12352.

Posted 1 month ago

Progress summary

Refreshed 1 month ago

Claimed solved

A February 2026 preprint claims an AI system proved the conjecture completely, but no independent confirmation has appeared.

Fel’s conjecture asserts a universal identity for the invariants $K_p(S)$ of every numerical semigroup and every $p \geq 0$ . Earlier work established the formula in low-degree cases and recorded the general expression conditionally.

Known results

Explicit identities were recorded through at least $K_6$ in earlier work, while the general formula was stated conditionally.
The catalogued literature contains two papers stating the conjecture, but neither earlier source reports a general proof or disproof.

February 2026 claimed proof

The preprint Fel’s Conjecture on Syzygies of Numerical Semigroups states that the identity holds for every $p \geq 0$ as Theorem 1.11, using exponential generating functions and coefficient extraction. It claims a Lean/Mathlib formalization produced automatically by AxiomProver; this is a claimed solution, not independently verified, and no correction, refutation, or referee confirmation was found.

Current status (as of September 2026): A general proof is claimed in the February 2026 preprint and attributed to AxiomProver, but its correctness and formal verification remain independently unconfirmed.

Sources

No solutions have been posted yet.