On factorization invariants and Hilbert functions

Abstract

Nonunique factorization in cancellative commutative semigroups is often studied using combinatorial factorization invariants, which assign to each semigroup element a quantity determined by the factorization structure. For numerical semigroups (additive subsemigroups of the natural numbers), several factorization invariants are known to admit predictable behavior for sufficiently large semigroup elements. In particular, the catenary degree and delta set invariants are both eventually periodic, and the omega-primality invariant is eventually quasilinear. In this paper, we demonstrate how each of these invariants is determined by Hilbert functions of graded modules. In doing so, we extend each of the aforementioned eventual behavior results to finitely generated semigroups, and provide a new framework through which to study factorization structures in this setting. arXiv:1503.08351v4 [math.AC] 23 Feb 2017 Definition 2.2. Fix a k-algebra S. An A-grading of S is an expression S ∼ = α∈A S α of S as a direct sum of finite dimensional k-subspaces of S, indexed by A, such that S α S β ⊂ S α+β for all α, β ∈ A. An A-grading of a module M over S is an expression M ∼ = α∈A

Factorization invariants in numerical monoids

“…More precisely, if S has k minimal generators, the size of Z(n) grows on the order of n k−1 . Theorem 2.6, which specializes the asymptotic result in [33] to numerical monoids, can be found in [38].…”

On length densities

“…Now, in order for LD(m i ) to be a strictly decreasing sequence, there must exist a gap size δ between sequential elements of L(m i ) satisfying LD(m i ) > 1/δ that occurs arbitrarily many times in L(m i ) for i large. Since M is finitely generated, the set of all trades with length difference at most δ is finite by [40,Theorem 4.9], so by omitting elements from the sequence m i as needed, we can assume there is some trade a ∼ b with |b| − |a| = δ between factorizations of some element b ∈ M with disjoint support that can be performed at least i times in Z(m i ) between factorizations of sequential length in L(m i ). This in particular implies b i | m i for all i.…”

numericalsgps, a GAP package for numerical semigroups

“…Along this line, we present methods for membership, computing minimal presentations, determine gluings, Betti and primitive elements, and the whole series of procedures for nonunique factorization invariants (an overview of the existing methods for the calculation of these invariants can be found in [30]). New procedures are now under development based in Hilbert functions and binomial ideals ( [37]). …”