Fel's Conjecture on Syzygies of Numerical Semigroups

Abstract: 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.

• The paper confirms Fel's conjecture by deriving a closed-form expression for the alternating power sums of syzygy degrees in numerical semigroups.
• It leverages exponential generating functions and universal symmetric polynomials Tₙ to connect syzygy invariants with gap power sums in semigroup rings.
• Explicit computations for binary and ternary cases validate the theoretical findings and demonstrate improved computational efficiency.

Resolution of Fel's Conjecture on Syzygies of Numerical Semigroups

Introduction and Background

This work addresses Fel's conjecture, which posits an explicit formula relating alternating power sums of syzygy degrees for numerical semigroups to gap power sums and a particular sequence of universal symmetric polynomials $T_n$. Numerical semigroups $S = \langle d_1, \ldots, d_m \rangle$ serve as a rich interface between additive number theory and commutative algebra. The syzygies of the semigroup ring $k[S]$ are classically encoded in the Hilbert series and its numerator $Q_S(z)$, a polynomial whose combinatorial structure reflects deep arithmetic properties of $S$. The degrees of syzygy—specifically, their alternating power sums—embed information about the generators and the gap set $\Delta = \mathbb{Z}_{\geq 0} \setminus S$.

The central object of study, the invariant $K_p(S)$, is defined as a normalized alternating sum of $(m+p)$-th powers of the syzygy degrees, yielding a surprising connection to the gap power sums $G_r(S) = \sum_{g \in \Delta} g^r$ and to a universal family $T_n$ of symmetric polynomials in the generator power sums $\sigma_k = \sum_i d_i^k$, and their shifted versions $\delta_k = (\sigma_k-1)/2^k$. Fel's conjecture proposed a closed-form expression for $K_p(S)$ in terms of these combinatorial and symmetric data for all $p \geq 0$.

Universal Symmetric Polynomials $T_n$ and Their Algebraic Emergence

The polynomials $T_n$ are established as fundamental symmetric invariants, defined through exponential generating functions as coefficients in expansions involving the exponential of the generators: $$A(t) = \prod_{i=1}^m \frac{e^{x_i t} - 1}{x_i t} \qquad T_n(x_1, \ldots, x_m) = n! \cdot [t^n] A(t)$$ Alternatively, expressed in terms of the power sums $\sigma_k$, these polynomials encapsulate universal relationships on the symmetric group invariants associated to the generators of $S$. They account for the coefficients that appear naturally in the expansion of the alternating syzygy power sums in terms of elementary combinatorial invariants of $S$.

Historically, the $T_n$ polynomials have appeared in diverse mathematical contexts—including the structure of restricted partition functions, Sylvester's waves, Ramanujan's $q$-series, and properties of quasimodular forms—demonstrating their canonical status beyond numerical semigroups. This cross-disciplinary reappearance enforces the theoretical unity underlying combinatorics, commutative algebra, and modular forms.

Main Theorem and Formal Proof

The main result rigorously confirms Fel's conjecture, proving that 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 identity bridges the gap between the algebraic syzygy invariants and combinatorial/arithmetic data from $S$ and $\Delta$. The proof employs a fully formalized argument based on exponential generating functions and term-by-term coefficient extraction.

Central steps include:

• Expressing $\CC_{m+p}(S)$, the alternating sum of syzygy degrees to the $(m+p)$th power, as $n!$ times the coefficient of $t^n$ in $1 - Q_S(e^t)$.
• Formulating both the gap polynomial $\Phi_S(z)$ and the Hilbert numerator $Q_S(z)$ as exponential generating functions, intertwining $G_r(S)$ and the $T_n$ polynomials via combinatorial convolution.
• Decomposing the structural polynomials $A(t)$ and $B(t)$ within this framework and extracting their contributions to the syzygy power sums.
• Achieving the explicit formula for $K_p(S)$ by equating corresponding coefficients and appropriately normalizing by $\pi_m$ and the relevant factorial weights.

The implementation in Lean/Mathlib via the AxiomProver system formalizes both the statement and proof of Fel's conjecture from a natural-language formulation, establishing mechanical verification for an area of mathematics traditionally subjected only to human proof scrutiny.

Numerical Examples and Explicit Computations

The paper provides explicit computations for small numerical semigroups, such as $S = \langle 3,5 \rangle$ and $S = \langle 4,5,6 \rangle$, verifying the formulae for $K_p(S)$ against directly computed syzygy and gap sums. For binary-generated semigroups, the syzygy structure reduces the Hilbert numerator to $1-z^{d_1 d_2}$, and $K_p(S)$ collapses to the expected closed-form $\frac{(d_1 d_2)^{p+1}}{(p+1)(p+2)}$. For ternary and higher cases, the explicit computation of $Q_S(z)$, the gap polynomial, and resulting $G_r(S)$ and $K_p(S)$ affirm the universality and concreteness of Fel's formula.

Practical and Theoretical Implications

The resolution of Fel's conjecture substantiates a deep link between arithmetic, algebraic invariants of semigroup rings, and universal symmetric polynomials. Practically, this formula provides an efficient mechanism to compute intricate syzygy invariants in terms of readily accessible data from $S$ and $\Delta$ without recourse to explicit minimal resolutions. Theoretically, it underlines the importance of symmetric function theory in understanding commutative algebraic structures and reveals that alternating syzygy invariants encode not just algebraic but also combinatorial and modular information.

The success of full formalization in Lean/Mathlib—driven by AxiomProver—signals ongoing advances in automated theorem proving for mainstream, research-level mathematics. It demonstrates that, at least for problems amenable to formal combinatorial and algebraic reasoning, natural-language conjectures can be fully mechanically verified. This suggests broader possibilities for rigorous, machine-assisted development in algebra, number theory, and related fields.

Conclusion

This work completes the resolution of long-standing conjectural relationships between the syzygies of numerical semigroups and explicit combinatorial invariants, fully characterizing the invariant $K_p(S)$ through universal symmetric polynomial and gap data. The result exemplifies a successful synthesis of commutative algebra, number theory, and symmetric polynomial theory, with significant implications for computational approaches to these questions. The formalization in Lean sets a precedent for machine-verified mathematics, strengthening the foundations of future research at the intersection of algebra, combinatorics, and automated reasoning.