Fel’s syzygy formulas for numerical semigroups

Can alternating power sums of the syzygy degrees of a numerical-semigroup ring be expressed by its gap power sums and universal symmetric polynomials? Fel’s conjecture gives an explicit normalized formula Kₚ(S) for every numerical semigroup S and integer p≥0.

Why it matters and what counts as progress

Why it matters: The identity connects homological degrees of a semigroup ring with the arithmetic of its missing integers.

What counts as progress: Prove the formula uniformly in S and p, give a complete formal derivation or extend the identities to further syzygy invariants.

How the problem developed

3 Nov 2017 · Foundational formulas [1]
Leonid G. Fel Derived polynomial and quasipolynomial identities for syzygy degrees of numerical-semigroup rings, providing background for later formulas involving gap power sums.
2022 · Conjecture [2]
Leonid G. Fel Formulated the remaining explicit formula for alternating syzygy power sums Kₚ(S), in terms of gap power sums and universal symmetric polynomials.
3 Feb 2026 · Formalized proof [3]
Axiom Math; AxiomProver generated and formalized the proof from a natural-language statement Proved Fel’s remaining formula for Kₚ(S) for every p ≥ 0 and supplied a Lean/Mathlib formalization.

AI contribution

Fel's Conjecture on Syzygies of Numerical Semigroups

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, Jujian Zhang

For every numerical semigroup S and integer p≥0, AxiomProver’s Lean proof establishes Fel’s formula for the normalized alternating syzygy quantity Kₚ(S) in terms of universal symmetric polynomials, gap power sums and a final polynomial term.

AI involvementAI-autonomous

ImportanceNotable

Technical verificationFormal (Lean 4/Mathlib kernel verification of the formal theorem)

Human understandingFull