Normalized Alternating Syzygy Power Sums

• Normalized alternating syzygy power sums are invariants in numerical semigroups that capture alternating sums of syzygy degrees through closed-form expressions involving universal symmetric polynomials.
• The methodology translates Hilbert series and syzygy data into exponential generating functions, enabling systematic extraction of combinatorial gap and generator power sums.
• These invariants provide explicit formulas critical for understanding minimal resolutions in semigroup rings, with broad applications in commutative algebra and algebraic geometry.

Normalized alternating syzygy power sums are invariants associated with numerical semigroups and their semigroup rings, encoding highly-structured information about the degrees of syzygies—central objects in commutative algebra and algebraic geometry. These invariants, denoted $K_p(S)$ for non-negative integers $p$, capture normalized alternating sums of powers of syzygy degrees, and admit explicit closed-form expressions in terms of the combinatorics of the semigroup’s generators and gap set, as well as universal symmetric polynomials. Recently, Fel's conjecture providing such a closed formula for all $K_p(S)$ was proved using exponential generating functions and algorithmic proof technology (Chen et al., 3 Feb 2026).

1. Numerical Semigroups and Semigroup Rings

A numerical semigroup $$S = \langle d_1, \dots, d_m \rangle \subset \mathbb{Z}$$ is the additive subsemigroup generated by distinct positive integers $d_1,\dots,d_m$ with $\gcd(d_1,\dots,d_m)=1$ and containing $0$. The finite gap set $\Delta = \mathbb{Z}\setminus S$ encodes the integers not in $S$. Associated is the semigroup ring $k[S] \cong k[t^{d_1},\dots,t^{d_m}]$, graded by $p$0, which is a standard object in the study of monomial curves and toric ideals.

The Hilbert series of $p$1 is given by

$p$2

with $p$3 the Hilbert numerator. This numerator is naturally expanded as

$p$4

where $p$5 are (partial) Betti numbers and $p$6 are syzygy-degrees.

2. Alternating Syzygy Power Sums and Their Normalization

The alternating syzygy power sum of order $p$7 for $p$8 is defined as

$p$9

Alternatively, writing $K_p(S)$0, one has $K_p(S)$1. With $K_p(S)$2, the normalized alternating syzygy power sums $K_p(S)$3 are implicitly defined by

$K_p(S)$4

or equivalently,

$K_p(S)$5

These invariants give a structured, dimensionally-normalized account of the power-sums of syzygy degrees with sign alternation according to syzygy rank.

3. Gap Power Sums, Generator Power Sums, and Universal Symmetric Polynomials

For a numerical semigroup $K_p(S)$6 generated by $K_p(S)$7:

• The gap power sums are $K_p(S)$8, capturing the $K_p(S)$9-th power sums over the set of gaps.
• The generator power sums are $$S = \langle d_1, \dots, d_m \rangle \subset \mathbb{Z}$$0 and $$S = \langle d_1, \dots, d_m \rangle \subset \mathbb{Z}$$1, encoding information about the generators' degrees.

Central to the closed-form description of $$S = \langle d_1, \dots, d_m \rangle \subset \mathbb{Z}$$2 are universal symmetric polynomials $$S = \langle d_1, \dots, d_m \rangle \subset \mathbb{Z}$$3, defined via the formal power series

$$S = \langle d_1, \dots, d_m \rangle \subset \mathbb{Z}$$4

and

$$S = \langle d_1, \dots, d_m \rangle \subset \mathbb{Z}$$5

The universal symmetric $$S = \langle d_1, \dots, d_m \rangle \subset \mathbb{Z}$$6 are symmetric of degree $$S = \langle d_1, \dots, d_m \rangle \subset \mathbb{Z}$$7 in the $$S = \langle d_1, \dots, d_m \rangle \subset \mathbb{Z}$$8 and are often expressed as $$S = \langle d_1, \dots, d_m \rangle \subset \mathbb{Z}$$9 in terms of the power-sum variables.

4. Main Formula and Proof via Exponential Generating Functions

Fel's conjecture, proved in (Chen et al., 3 Feb 2026), states that for every $d_1,\dots,d_m$0: $d_1,\dots,d_m$1 Thus, $d_1,\dots,d_m$2 is a universal linear combination of the gap power sums $d_1,\dots,d_m$3, with coefficients given by the $d_1,\dots,d_m$4-polynomials evaluated on generator power sums and associated terms involving $d_1,\dots,d_m$5. The proof synthesizes alternating sum expressions into exponential generating functions, enabling use of generating function identities and systematic coefficient extraction.

Key steps are:

• Translating the alternating sum $d_1,\dots,d_m$6 and $d_1,\dots,d_m$7 into their exponential generating functions.
• Factoring denominators using properties of the $d_1,\dots,d_m$8 and connecting with $d_1,\dots,d_m$9.
• Substituting $\gcd(d_1,\dots,d_m)=1$0 and equating coefficients of $\gcd(d_1,\dots,d_m)=1$1 to solve for $\gcd(d_1,\dots,d_m)=1$2.

These derivations were formalized in Lean/Mathlib and computationally verified by AxiomProver.

5. Explicit Computations and Illustrative Examples

The formula for $\gcd(d_1,\dots,d_m)=1$3 facilitates explicit calculations in practical cases. Selected examples include:

$\gcd(d_1,\dots,d_m)=1$4 generators	$\gcd(d_1,\dots,d_m)=1$5 (gaps)	$\gcd(d_1,\dots,d_m)=1$6
$\gcd(d_1,\dots,d_m)=1$7	$\gcd(d_1,\dots,d_m)=1$8	$\gcd(d_1,\dots,d_m)=1$9
$0$0	$0$1	$0$2
$0$3	(as per full data)	As detailed in (Chen et al., 3 Feb 2026), via sums/differences of powers

For $0$4, $0$5 thus $0$6 so

$0$7

in agreement with the universal formula. For $0$8 with $0$9, one has

$\Delta = \mathbb{Z}\setminus S$0

The four-generator case exhibits similar structure but with more elaborate alternating sum terms.

6. Methodological Significance and Formal Verification

The proof strategy for the closed-form of $\Delta = \mathbb{Z}\setminus S$1 leverages the translation of Hilbert numerator and syzygy degree information into exponential generating functions, facilitating algorithmic expansion and coefficient extraction. Fundamental identities between $\Delta = \mathbb{Z}\setminus S$2 and $\Delta = \mathbb{Z}\setminus S$3 allow expression of all relevant quantities in terms of universal symmetric polynomials and combinatorial data from $\Delta = \mathbb{Z}\setminus S$4. The argument is formalized in the Lean/Mathlib formal proof system and was automatically generated via AxiomProver from a natural-language conjecture, illustrating the integration of symbolic computation and automated mathematical proof in modern algebraic research.

7. Connections and Broader Context

Normalized alternating syzygy power sums bridge the algebraic invariants of numerical semigroups, combinatorial gap statistics, and structure theory of semigroup rings. The main formula highlights the universality of the relationship, in which gap power sums and generator power sums are combined through symmetric polynomials to reflect deep properties of the ring’s minimal resolution. This connection supports further advances in the explicit study of Betti numbers, syzygy degrees, and more general phenomena in the homological algebra of monomial semigroup rings. The computational methodology and formal verification signal robust avenues for extending such explicit algebraic formulae in related domains (Chen et al., 3 Feb 2026).