Maximal Denumerant of a Numerical Semigroup With Embedding Dimension Less Than Four

Given a numerical semigroup $S =$ and $s\in S$, we consider the factorization $s = c_1 a_1 + c_2 a_2 +... + c_t a_t$ where $c_i\ge0$. Such a factorization is {\em maximal} if $c_1+c_2+...+c_t$ is a maximum over all such factorizations of $s$. We show that the number of maximal factorizations, varying over the elements in $S$, is always bounded. Thus, we define \dx(S) to be the maximum number of maximal factorizations of elements in $S$. We study maximal factorizations in depth when $S$ has embedding dimension less than four, and establish formulas for \dx(S) in this case.Comment: Main results are unchanged, but proofs and exposition have been improved. Some details have been changed considerably including the titl

The Maximal Denumerant of a Numerical Semigroup

Given a numerical semigroup S = and n in S, we consider the factorization n = c_0 a_0 + c_1 a_1 + ... + c_t a_t where c_i >= 0. Such a factorization is maximal if c_0 + c_1 + ... + c_t is a maximum over all such factorizations of n. We provide an algorithm for computing the maximum number of maximal factorizations possible for an element in S, which is called the maximal denumerant of S. We also consider various cases that have connections to the Cohen-Macualay and Gorenstein properties of associated graded rings for which this algorithm simplifies.Comment: 13 Page

Factorization and catenary degree in 3-generated numerical semigroups

Given a numerical semigroup S(A), generated by A = {a,b,N} ⊂ N with 0 < a < b < N and gcd(a,b,N) = 1, we give a parameterization of the set F(m;A) = {(x, y, z) ∈ $N^3$ | xa + yb + zN = m} for any m ∈ S(A). We also give the catenary degree of S(A), c(A). Boths results need the computation of an L-shaped tile, related to the set A, that has time-complexity O(logN) in the worst case.Postprint (published version

Factorization and catenary degree in 3-generated numerical semigroups

Given a numerical semigroup S(A), generated by A = {a,b,N} ⊂ N with 0 < a < b < N and gcd(a,b,N) = 1, we give a parameterization of the set F(m;A) = {(x, y, z) ∈ $N^3$ | xa + yb + zN = m} for any m ∈ S(A). We also give the catenary degree of S(A), c(A). Boths results need the computation of an L-shaped tile, related to the set A, that has time-complexity O(logN) in the worst case