Fel's Conjecture on Syzygies of Numerical Semigroups

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.

💡 Research Summary

The paper addresses a conjecture made by Fel concerning numerical semigroups and the alternating power‑sum invariants that arise from the Hilbert numerator of the semigroup ring. For a numerical semigroup (S=\langle d_1,\dots ,d_m\rangle) the Hilbert series (H_S(z)=\sum_{s\in S}z^s) can be written as a rational function whose numerator (Q_S(z)) is an alternating sum of monomials involving the partial Betti numbers (\beta_i) and the syzygy degrees (C_{i,j}). From these data Fel defined the alternating power sums \