Fel's Conjecture on Syzygies

• Fel's Conjecture on Syzygies is a theoretical framework that defines closed formulas for normalized syzygy power sums in numerical semigroup rings.
• It employs exponential generating functions and combinatorial identities to connect invariants like gap power sums and generator power sums.
• The conjecture offers structural insights into Betti numbers and paves the way for automated formalization in advanced commutative algebra.

@@@@1@@@@ on Syzygies concerns the alternating power sums of syzygy degrees in the graded minimal free resolution of numerical semigroup rings. This conjecture posits explicit closed formulas for normalized syzygy power sums, using invariants associated to the numerical semigroup—particularly the power sums over its set of "gaps" and certain universal symmetric polynomials evaluated at the generators. The conjecture reflects deep structural properties of Betti degrees and connects homological, combinatorial, and symmetric function perspectives in the study of semigroup rings (Chen et al., 3 Feb 2026, Martino et al., 2011).

1. Numerical Semigroups, Semigroup Rings, and Syzygies

A numerical semigroup $S = \langle d_1, \dots, d_m \rangle$ is the additive submonoid of $\mathbb N$ generated by $d_1, \dots, d_m \in \mathbb N$ with $\gcd(d_1, \dots, d_m) = 1$. The finite complement $\Delta = \mathbb N \setminus S$ is called the gap set; its cardinality is the genus. The semigroup ring $k[S]$ over a field $k$ possesses a natural grading, with $\deg t^{d_i} = d_i$. Its structure is captured homologically by its minimal graded free resolution over $R = k[x_1, ..., x_m]$, where $\deg x_i = d_i$. This resolution provides Betti numbers $\beta_{i,j}$, encoding the number of $i$th syzygies of degree $j$, and Betti sums $\beta_i = \sum_j \beta_{i,j}$.

The Hilbert series of $k[S]$ is given by

$$H_S(z) = \sum_{s \in S} z^s = \frac{Q_S(z)}{\prod_{i=1}^m (1 - z^{d_i})},$$

where $Q_S(z) \in \mathbb Z[z]$ is the Hilbert numerator. The expansion of $Q_S(z)$ allows the syzygy degrees to be read as exponents, providing a bridge between combinatorial, algebraic, and homological perspectives (Chen et al., 3 Feb 2026, Martino et al., 2011).

2. Alternating Power Sums and Normalized Syzygy Power Sums

The characteristic invariants of interest are the alternating power sums of the syzygy degrees, defined for $r \ge 0$ as

$$C_r(S) = \sum_{i=1}^{m-1} \sum_{j=1}^{\beta_i} (-1)^i C_{i,j}^r,$$

where $C_{i,j}$ are the degrees of the $j$th $i$th syzygy. Fel proved the vanishing of these sums for $1 \le r \le m-2$ and provided an explicit non-vanishing formula for $r = m-1$:

$C_{m-1}(S) = (-1)^m (m-1)! \prod_{i=1}^m d_i.$

The normalization

$$C_{m+p}(S) = (-1)^m \pi_m \cdot \frac{(m+p)!}{p!} \cdot K_p(S), \quad \pi_m = \prod_{i=1}^m d_i$$

defines the normalized syzygy power sums $K_p(S)$. These quantities encode the combinatorial structure of syzygy degrees and serve as the central objects in Fel's conjecture (Chen et al., 3 Feb 2026, Martino et al., 2011).

3. Statement and Interpretation of Fel's Conjecture

Fel's Conjecture asserts that, for all $p \ge 0$,

$$K_p(S) = \sum_{r=0}^p \binom{p}{r}\,T_{p-r}(\sigma_1, \dots, \sigma_{p-r})\, G_r(S)\,+\,\frac{2^{p+1}}{p+1} T_{p+1}(\delta_1, \dots, \delta_{p+1}),$$

where:

• $G_r(S) = \sum_{g \in \Delta} g^r$ is the $r$th gap power sum,
• $\sigma_k = \sum_{i=1}^m d_i^k$ are generator power sums,
• $\delta_k = (\sigma_k - 1)/2^k$ serves as a shifted-normalization,
• $T_n$ are universal symmetric polynomials in the power-sum basis (arising as coefficients in the expansion of

$$A(t) = \prod_{i=1}^m \frac{e^{x_i t} - 1}{x_i t} = \sum_{n \ge 0} T_n(x_1, \dots, x_m) \frac{t^n}{n!}$$

and their normalization appears through the modified generating function $B(t) = \frac{t}{e^t - 1} A(t)$).

The conjecture explicitly expresses the higher-alternating power sums of syzygy degrees as a combination of the gap structure and the algebraic data of the generators, mediated by universal symmetric polynomials (Chen et al., 3 Feb 2026).

4. Proof Techniques and Formalization

The proof of Fel's Conjecture utilizes exponential generating function (EGF) techniques combined with coefficient extraction. The Hilbert numerator $Q_S(z)$, the set of gaps, and the generator sequence are encoded into exponential generating functions in a formal parameter $t$. Specifically:

• Substitution $z \mapsto e^t$ translates Hilbert series manipulations to coefficient comparisons in EGFs.
• The gap polynomial $\Phi_S(z)$ and its EGF expansion provide $G_r(S)$ as coefficients.
• The denominator $\prod_{i=1}^m (1 - z^{d_i})$ is encoded via the function $A(t)$, and its variant $B(t)$, both with coefficients $T_n(\sigma_1, \ldots, \sigma_n)$ and $T_n(\delta_1, \ldots, \delta_n)$.
• The identity $$1 - Q_S(z) = 1 - \frac{P_S(z)}{1-z} + \Phi_S(z) P_S(z)$$, after substituting and expanding at $z = e^t$, yields—by comparing coefficients—the explicit formula for $K_p(S)$.

The entire argument is formalized in Lean 4/Mathlib, derived automatically with the AxiomProver system from a natural-language conjecture. This formalization confirms both the soundness of the proof and the general strategy, offering a proof-of-concept for automated formalization of advanced commutative algebra results (Chen et al., 3 Feb 2026).

5. Relevant Identities and Generalizations

Fel's identities for syzygy degrees were initially discovered for small powers; specifically, for $0 \leq r \leq m-2$:

$$\sum_{i=0}^{m-1} \sum_j (-1)^i \beta_{i,j} j^r = 0,$$

with a unique nontrivial case for $r = m-1$:

$$\sum_{i=0}^{m-1} \sum_j (-1)^i \beta_{i,j} j^{m-1} = (-1)^{m-1} (m-1)! \prod_{i=1}^m d_i.$$

These relations were given algebraic and homological proofs relying only on the behavior of the Hilbert series, derivatives, and elementary properties of polynomials with high-order roots at $z = 1$ (Martino et al., 2011).

Complex-valued generalizations are obtained by evaluating the same alternating sums with multiplicative twists $\exp(2\pi i n j/q)$ at roots of unity, yielding higher-multiplicity vanishing relations under arithmetic divisibility constraints among the generators (Martino et al., 2011). This demonstrates a robust structural vanishing over both the reals and the complex field.

6. Examples and Concrete Computations

Illustrative applications confirm the general formulae:

• $S = \langle 3,5 \rangle$ yields $\Delta = \{1,2,4,7\}$, $G_r(S) = 1+2^r+4^r+7^r$, $\sigma_k = 3^k+5^k$, and $\delta_k = (3^k+5^k-1)/2^k$. The computation shows $K_p(S) = 15^{p+1}/[(p+1)(p+2)]$, in agreement with Fel's formula.
• $S = \langle 4,5,6 \rangle$ gives $\Delta = \{1,2,3,7\}$, with $\sigma_k = 4^k+5^k+6^k$ and $\delta_k = (\sigma_k-1)/2^k$. The result $$K_p(S) = [22^{p+3} - 10^{p+3} - 12^{p+3}] / [120(p+1)(p+2)(p+3)]$$ matches the conjectural prediction (Chen et al., 3 Feb 2026).

These calculations exemplify the exactness and generality of Fel's formula for semigroups with differing numbers and arrangements of generators.

7. Open Questions and Extensions

Several open directions stem from Fel's Conjecture:

• A combinatorial understanding of the normalized syzygy power sums, in terms of partitions or lattice polytopes corresponding to the semigroup.
• The possibility of extending the identities and formulas to higher-dimensional affine semigroups or to more general Cohen–Macaulay monoid rings.
• The use of these results to bound or characterize the Betti numbers $\beta_{i,j}$ and their locus within the degree-syzygy index plane.

A plausible implication is that Fel's structure theorems might inform or inspire approaches to long-standing problems in the study of syzygies, Betti tables, and their extremal behavior for more general algebraic structures (Chen et al., 3 Feb 2026, Martino et al., 2011).