Sylvester Sums: Theory and Applications

• Sylvester sums are power sums over nonrepresentable integers in numerical semigroups, central to Frobenius problems and gap analysis.
• They generalize to weighted and higher-power variants through generating-function approaches and recursive combinatorial methods.
• Sylvester double sums appear in elimination theory, linking root-symmetric expressions with subresultants and interpolation techniques.

Sylvester sums are a family of constructions named after J. J. Sylvester that occur in several adjacent literatures. In numerical semigroup theory and the Frobenius problem, they are the power sums of the nonrepresentable positive integers; in elimination theory they are the root-symmetric single and double sums that encode resultants and subresultants; and in partition theory the related language of Sylvester waves refers to quasiperiodic components of restricted partition functions. The dominant modern usage concerns the Frobenius set $\operatorname{NR}(a_1,\dots,a_k)$, where the basic Sylvester sum is $s(a_1,\dots,a_k)=\sum_{n\in\operatorname{NR}} n$, but contemporary work also studies higher powers, weighted variants, structured multi-generator families, and the classical double-sum formulas attached to two polynomials (Komatsu, 2022, Krick et al., 2015, Rubinstein, 23 Dec 2025).

1. Numerical semigroups, gaps, and the basic Sylvester invariants

Let $a_1,\dots,a_k$ be positive integers with $\gcd(a_1,\dots,a_k)=1$. The numerical semigroup they generate is

$$\langle a_1,\dots,a_k\rangle=\Big\{\sum_{i=1}^k x_i a_i : x_i\in\mathbb{Z}_{\ge 0}\Big\}.$$

The set of positive integers that are not representable in this form is

$$\operatorname{NR}(a_1,\dots,a_k)=\{n\in\mathbb{Z}_{>0}:n\notin\langle a_1,\dots,a_k\rangle\}.$$

Within this framework, the Frobenius number is the largest gap,

$$g(a_1,\dots,a_k)=\max \operatorname{NR}(a_1,\dots,a_k),$$

the Sylvester number is the cardinality

$$n(a_1,\dots,a_k)=\#\operatorname{NR}(a_1,\dots,a_k),$$

and the Sylvester sum is

$$s(a_1,\dots,a_k)=\sum_{n\in \operatorname{NR}(a_1,\dots,a_k)} n.$$

In numerical semigroup theory, the Sylvester number is also the genus (Komatsu, 2022).

For two generators, the classical theory is explicit. If $\gcd(a,b)=1$, then

$s(a_1,\dots,a_k)=\sum_{n\in\operatorname{NR}} n$0

and Brown–Shiue’s formula gives

$s(a_1,\dots,a_k)=\sum_{n\in\operatorname{NR}} n$1

The higher-power sums

$s(a_1,\dots,a_k)=\sum_{n\in\operatorname{NR}} n$2

are also called Sylvester sums in the two-generator literature. In particular,

$s(a_1,\dots,a_k)=\sum_{n\in\operatorname{NR}} n$3

is one of the classical explicit formulas recalled in the recent weighted-sum literature (Komatsu et al., 2021).

2. Two-generator theory: power sums, weights, and recursion

For coprime $s(a_1,\dots,a_k)=\sum_{n\in\operatorname{NR}} n$4, one standard notation is

$s(a_1,\dots,a_k)=\sum_{n\in\operatorname{NR}} n$5

so that $s(a_1,\dots,a_k)=\sum_{n\in\operatorname{NR}} n$6 and $s(a_1,\dots,a_k)=\sum_{n\in\operatorname{NR}} n$7. Recent work has enlarged this to weighted Sylvester sums

$s(a_1,\dots,a_k)=\sum_{n\in\operatorname{NR}} n$8

with $s(a_1,\dots,a_k)=\sum_{n\in\operatorname{NR}} n$9. The cases $a_1,\dots,a_k$0 and $a_1,\dots,a_k$1 recover, respectively, the classical Sylvester sums and the alternating Sylvester sums $a_1,\dots,a_k$2 (Komatsu et al., 2021).

The generating-function treatment of these weighted sums factors a polynomial

$a_1,\dots,a_k$3

as $a_1,\dots,a_k$4, where $a_1,\dots,a_k$5 is the number of representations of $a_1,\dots,a_k$6 as $a_1,\dots,a_k$7 with $a_1,\dots,a_k$8. Differentiation then identifies $a_1,\dots,a_k$9 with $\gcd(a_1,\dots,a_k)=1$0, and higher derivatives yield $\gcd(a_1,\dots,a_k)=1$1. For general $\gcd(a_1,\dots,a_k)=1$2, the resulting formulas are expressed in terms of Apostol–Bernoulli numbers; the singular case $\gcd(a_1,\dots,a_k)=1$3 is treated separately and leads back to classical Bernoulli numbers (Komatsu et al., 2021).

A second modern approach is recursive and combinatorial. Every $\gcd(a_1,\dots,a_k)=1$4 with $\gcd(a_1,\dots,a_k)=1$5 can be written uniquely as

$\gcd(a_1,\dots,a_k)=1$6

with $\gcd(a_1,\dots,a_k)=1$7, $\gcd(a_1,\dots,a_k)=1$8, where $\gcd(a_1,\dots,a_k)=1$9 in the nonrepresentable case and $$\langle a_1,\dots,a_k\rangle=\Big\{\sum_{i=1}^k x_i a_i : x_i\in\mathbb{Z}_{\ge 0}\Big\}.$$0 in the representable case. Summing powers of $$\langle a_1,\dots,a_k\rangle=\Big\{\sum_{i=1}^k x_i a_i : x_i\in\mathbb{Z}_{\ge 0}\Big\}.$$1 in two ways yields a recursive formula for the Sylvester sums $$\langle a_1,\dots,a_k\rangle=\Big\{\sum_{i=1}^k x_i a_i : x_i\in\mathbb{Z}_{\ge 0}\Big\}.$$2, and the same framework gives a criterion for deciding whether a given integer is representable as $$\langle a_1,\dots,a_k\rangle=\Big\{\sum_{i=1}^k x_i a_i : x_i\in\mathbb{Z}_{\ge 0}\Big\}.$$3 (Gupta et al., 9 Jul 2025).

These two lines of work are complementary. The generating-function/Apostol–Bernoulli approach produces explicit weighted formulas uniform in $$\langle a_1,\dots,a_k\rangle=\Big\{\sum_{i=1}^k x_i a_i : x_i\in\mathbb{Z}_{\ge 0}\Big\}.$$4, while the recursive approach derives the unweighted $$\langle a_1,\dots,a_k\rangle=\Big\{\sum_{i=1}^k x_i a_i : x_i\in\mathbb{Z}_{\ge 0}\Big\}.$$5 from a direct combinatorial decomposition of the interval $$\langle a_1,\dots,a_k\rangle=\Big\{\sum_{i=1}^k x_i a_i : x_i\in\mathbb{Z}_{\ge 0}\Big\}.$$6. This suggests a durable bifurcation in the modern theory between analytic and combinatorial normal forms.

3. Multi-generator generalizations and Apéry-set formulas

When $$\langle a_1,\dots,a_k\rangle=\Big\{\sum_{i=1}^k x_i a_i : x_i\in\mathbb{Z}_{\ge 0}\Big\}.$$7, closed formulas are rare and typically exist only for special structured families. The general computational framework is the Apéry set with respect to the least generator $$\langle a_1,\dots,a_k\rangle=\Big\{\sum_{i=1}^k x_i a_i : x_i\in\mathbb{Z}_{\ge 0}\Big\}.$$8: $$\langle a_1,\dots,a_k\rangle=\Big\{\sum_{i=1}^k x_i a_i : x_i\in\mathbb{Z}_{\ge 0}\Big\}.$$9 where $$\operatorname{NR}(a_1,\dots,a_k)=\{n\in\mathbb{Z}_{>0}:n\notin\langle a_1,\dots,a_k\rangle\}.$$0 is the least nonnegative representable integer congruent to $$\operatorname{NR}(a_1,\dots,a_k)=\{n\in\mathbb{Z}_{>0}:n\notin\langle a_1,\dots,a_k\rangle\}.$$1, with $$\operatorname{NR}(a_1,\dots,a_k)=\{n\in\mathbb{Z}_{>0}:n\notin\langle a_1,\dots,a_k\rangle\}.$$2. Standard formulas express the Frobenius number, Sylvester number, and Sylvester sum as

$$\operatorname{NR}(a_1,\dots,a_k)=\{n\in\mathbb{Z}_{>0}:n\notin\langle a_1,\dots,a_k\rangle\}.$$3

$$\operatorname{NR}(a_1,\dots,a_k)=\{n\in\mathbb{Z}_{>0}:n\notin\langle a_1,\dots,a_k\rangle\}.$$4

$$\operatorname{NR}(a_1,\dots,a_k)=\{n\in\mathbb{Z}_{>0}:n\notin\langle a_1,\dots,a_k\rangle\}.$$5

For structured families, the main task is therefore to determine $$\operatorname{NR}(a_1,\dots,a_k)=\{n\in\mathbb{Z}_{>0}:n\notin\langle a_1,\dots,a_k\rangle\}.$$6 explicitly and then compute $$\operatorname{NR}(a_1,\dots,a_k)=\{n\in\mathbb{Z}_{>0}:n\notin\langle a_1,\dots,a_k\rangle\}.$$7 and $$\operatorname{NR}(a_1,\dots,a_k)=\{n\in\mathbb{Z}_{>0}:n\notin\langle a_1,\dots,a_k\rangle\}.$$8 (Komatsu, 2022).

Weighted sums in more variables admit a parallel Apéry-set description. If

$$\operatorname{NR}(a_1,\dots,a_k)=\{n\in\mathbb{Z}_{>0}:n\notin\langle a_1,\dots,a_k\rangle\}.$$9

then the decomposition of $$g(a_1,\dots,a_k)=\max \operatorname{NR}(a_1,\dots,a_k),$$0 into arithmetic progressions inside each congruence class modulo $$g(a_1,\dots,a_k)=\max \operatorname{NR}(a_1,\dots,a_k),$$1 reduces the problem to finite exponential–polynomial sums. In this setting the coefficients are expressed with Eulerian numbers when $$g(a_1,\dots,a_k)=\max \operatorname{NR}(a_1,\dots,a_k),$$2, while Bernoulli numbers govern the unweighted power sums $$g(a_1,\dots,a_k)=\max \operatorname{NR}(a_1,\dots,a_k),$$3 (Komatsu et al., 2021).

Another important structured class is given by compound sequence semigroups. If $$g(a_1,\dots,a_k)=\max \operatorname{NR}(a_1,\dots,a_k),$$4 is a compound sequence arising from a suitable pair $$g(a_1,\dots,a_k)=\max \operatorname{NR}(a_1,\dots,a_k),$$5, then the complement $$g(a_1,\dots,a_k)=\max \operatorname{NR}(a_1,\dots,a_k),$$6 satisfies a generalized Tuenter identity, and the semigroup is symmetric. In particular,

$$g(a_1,\dots,a_k)=\max \operatorname{NR}(a_1,\dots,a_k),$$7

and explicit formulas are given for $$g(a_1,\dots,a_k)=\max \operatorname{NR}(a_1,\dots,a_k),$$8, $$g(a_1,\dots,a_k)=\max \operatorname{NR}(a_1,\dots,a_k),$$9, and $$n(a_1,\dots,a_k)=\#\operatorname{NR}(a_1,\dots,a_k),$$0, together with a Bernoulli-number formula for general $$n(a_1,\dots,a_k)=\#\operatorname{NR}(a_1,\dots,a_k),$$1 (Gassert et al., 2016).

4. Explicit formulas for structured families

Arithmetic progressions and almost arithmetic sequences form the most developed class of closed-form examples. For generators

$$n(a_1,\dots,a_k)=\#\operatorname{NR}(a_1,\dots,a_k),$$2

explicit formulas are known for the simple Sylvester sum $$n(a_1,\dots,a_k)=\#\operatorname{NR}(a_1,\dots,a_k),$$3 and for the weighted sum

$$n(a_1,\dots,a_k)=\#\operatorname{NR}(a_1,\dots,a_k),$$4

The same methods extend to almost arithmetic sequences $$n(a_1,\dots,a_k)=\#\operatorname{NR}(a_1,\dots,a_k),$$5, to arithmetic sequences with an additional term $$n(a_1,\dots,a_k)=\#\operatorname{NR}(a_1,\dots,a_k),$$6, and to a geometric-like sequence $$n(a_1,\dots,a_k)=\#\operatorname{NR}(a_1,\dots,a_k),$$7 (Komatsu, 2022).

A particularly rich extension is the family with an initial block of missing terms,

$$n(a_1,\dots,a_k)=\#\operatorname{NR}(a_1,\dots,a_k),$$8

where $$n(a_1,\dots,a_k)=\#\operatorname{NR}(a_1,\dots,a_k),$$9 are omitted. Writing

$$s(a_1,\dots,a_k)=\sum_{n\in \operatorname{NR}(a_1,\dots,a_k)} n.$$0

the formulas for the Apéry set, Frobenius number, Sylvester number, and Sylvester sum split according to the size of $$s(a_1,\dots,a_k)=\sum_{n\in \operatorname{NR}(a_1,\dots,a_k)} n.$$1. In the small-gap regime $$s(a_1,\dots,a_k)=\sum_{n\in \operatorname{NR}(a_1,\dots,a_k)} n.$$2, one has

$$s(a_1,\dots,a_k)=\sum_{n\in \operatorname{NR}(a_1,\dots,a_k)} n.$$3

while in the bigger-gap regime $$s(a_1,\dots,a_k)=\sum_{n\in \operatorname{NR}(a_1,\dots,a_k)} n.$$4 the Frobenius number acquires additional case distinctions in $$s(a_1,\dots,a_k)=\sum_{n\in \operatorname{NR}(a_1,\dots,a_k)} n.$$5 and an extra $$s(a_1,\dots,a_k)=\sum_{n\in \operatorname{NR}(a_1,\dots,a_k)} n.$$6-type shift (Komatsu, 2022).

Several special four-generator families collapse to compact floor formulas. For

$$s(a_1,\dots,a_k)=\sum_{n\in \operatorname{NR}(a_1,\dots,a_k)} n.$$7

$$s(a_1,\dots,a_k)=\sum_{n\in \operatorname{NR}(a_1,\dots,a_k)} n.$$8

and for

$$s(a_1,\dots,a_k)=\sum_{n\in \operatorname{NR}(a_1,\dots,a_k)} n.$$9

$\gcd(a,b)=1$0

The corresponding Sylvester sums and Sylvester numbers are given by explicit piecewise polynomials in $\gcd(a,b)=1$1 modulo $\gcd(a,b)=1$2 or $\gcd(a,b)=1$3; analogous but more involved formulas exist for $\gcd(a,b)=1$4 and $\gcd(a,b)=1$5 (Komatsu, 2022).

These families show that the obstruction to closed formulas is not the absence of algebraic structure but the combinatorial complexity of the Apéry set. A plausible implication is that the central difficulty lies in residue-class geometry rather than in the definitions of $\gcd(a,b)=1$6, $\gcd(a,b)=1$7, and $\gcd(a,b)=1$8 themselves.

5. Sylvester double sums and elimination theory

A second major meaning of Sylvester sums belongs to elimination theory. Let

$\gcd(a,b)=1$9

and for finite sets $s(a_1,\dots,a_k)=\sum_{n\in\operatorname{NR}} n$00 write

$s(a_1,\dots,a_k)=\sum_{n\in\operatorname{NR}} n$01

Sylvester’s double sums are the root-symmetric polynomials indexed by $s(a_1,\dots,a_k)=\sum_{n\in\operatorname{NR}} n$02, $s(a_1,\dots,a_k)=\sum_{n\in\operatorname{NR}} n$03, built from subsets $s(a_1,\dots,a_k)=\sum_{n\in\operatorname{NR}} n$04, $s(a_1,\dots,a_k)=\sum_{n\in\operatorname{NR}} n$05 and the products $s(a_1,\dots,a_k)=\sum_{n\in\operatorname{NR}} n$06, $s(a_1,\dots,a_k)=\sum_{n\in\operatorname{NR}} n$07, $s(a_1,\dots,a_k)=\sum_{n\in\operatorname{NR}} n$08, and $s(a_1,\dots,a_k)=\sum_{n\in\operatorname{NR}} n$09. Their degree in $s(a_1,\dots,a_k)=\sum_{n\in\operatorname{NR}} n$10 is at most $s(a_1,\dots,a_k)=\sum_{n\in\operatorname{NR}} n$11, and they satisfy the symmetry

$s(a_1,\dots,a_k)=\sum_{n\in\operatorname{NR}} n$12

in the simple-root setting (Krick et al., 2015).

The fundamental structural fact is that these double sums are equivalent to subresultants and Bézout coefficients. If $s(a_1,\dots,a_k)=\sum_{n\in\operatorname{NR}} n$13, then

$s(a_1,\dots,a_k)=\sum_{n\in\operatorname{NR}} n$14

For larger $s(a_1,\dots,a_k)=\sum_{n\in\operatorname{NR}} n$15, the remaining cases are either zero or linear combinations of a subresultant and a Bézout coefficient times $s(a_1,\dots,a_k)=\sum_{n\in\operatorname{NR}} n$16 or $s(a_1,\dots,a_k)=\sum_{n\in\operatorname{NR}} n$17. Symmetric multivariate Lagrange interpolation and the Exchange Lemma provide a short conceptual route to these identities (Krick et al., 2015).

With multiple roots, the original quotient-of-differences formula no longer makes sense because denominators vanish. A generalized definition uses antisymmetrization, generalized Vandermonde determinants, and symmetric multivariate Hermite interpolation. In that setting, double sums still depend, up to an explicit constant, only on $s(a_1,\dots,a_k)=\sum_{n\in\operatorname{NR}} n$18, and they coincide with subresultants up to an explicit scalar factor (Roy et al., 2018).

This elimination-theoretic meaning of Sylvester sums is operational in rational interpolation. In the Cauchy interpolation problem, subresultants and their Bézout coefficients give the interpolant $s(a_1,\dots,a_k)=\sum_{n\in\operatorname{NR}} n$19, and Sylvester single-sum formulas yield explicit root-symmetric expressions for the numerator and denominator. In the osculatory case, where multiplicities intervene, determinantal formulas replace the unavailable general root formulas (D'Andrea et al., 2012).

6. Related notions, applications, and adjacent uses

The term Sylvester also appears in the theory of denumerants and restricted partitions. For a tuple $s(a_1,\dots,a_k)=\sum_{n\in\operatorname{NR}} n$20, the Sylvester denumerant

$s(a_1,\dots,a_k)=\sum_{n\in\operatorname{NR}} n$21

is a quasi-polynomial in $s(a_1,\dots,a_k)=\sum_{n\in\operatorname{NR}} n$22, and Sylvester decomposed it into waves $s(a_1,\dots,a_k)=\sum_{n\in\operatorname{NR}} n$23, one for each relevant root-of-unity frequency. Recent work derives explicit formulas for these waves by $s(a_1,\dots,a_k)=\sum_{n\in\operatorname{NR}} n$24-partial fractions, reciprocal degenerate Bernoulli numbers, and generalized Fourier–Dedekind sums (Kiran, 2021).

A parallel line studies the restricted partition function

$s(a_1,\dots,a_k)=\sum_{n\in\operatorname{NR}} n$25

which Sylvester decomposed as

$s(a_1,\dots,a_k)=\sum_{n\in\operatorname{NR}} n$26

Here $s(a_1,\dots,a_k)=\sum_{n\in\operatorname{NR}} n$27 is the polynomial part and the remaining $s(a_1,\dots,a_k)=\sum_{n\in\operatorname{NR}} n$28 are quasiperiodic components called Sylvester waves. Explicit expressions are now available as finite sums over Bernoulli polynomials of higher order with periodic coefficients, and also as weighted sums of polynomial terms with shifted arguments (Rubinstein, 23 Dec 2025).

In algebraic geometry, Sylvester sums of compound sequence semigroup complements determine the genus of certain towers of superelliptic curves and the $s(a_1,\dots,a_k)=\sum_{n\in\operatorname{NR}} n$29-Weierstrass weight of the unique point at infinity. In particular, if $s(a_1,\dots,a_k)=\sum_{n\in\operatorname{NR}} n$30 is that point, then

$s(a_1,\dots,a_k)=\sum_{n\in\operatorname{NR}} n$31

so a Frobenius-set invariant controls higher-order Weierstrass data (Gassert et al., 2016).

There are also categorical and topological reinterpretations. In the foam-based representation-theoretic setting of iterated wreath products, overlapping foams are used to interpret functors and natural transformations, and the same framework explores a relation between overlapping foams and Sylvester double sums (Im et al., 2021). This suggests that the term now names not a single invariant but a network of closely related constructions: gap sums in numerical semigroups, subset-symmetric elimination formulas, and quasiperiodic components of partition enumerators.