Numerical Semigroups Research Articles

On the type and generators of monomial curves

Let $n_1, n_2,\ldots, n_d$ be positive integers and $H $ be the numerical semigroup generated by $n_1,n_2, \ldots, n_d$. Let $A:=k[H]:=k[t^{n_1}, t^{n_2},\ldots, t^{n_d}]\cong k[x_1,x_2,\ldots,x_d]/I$ be the numerical semigroup ring of $H $ over $k.$ In this paper we give a condition $(*)$ that implies that the minimal number of generators of the defining ideal $I$ is bounded explicitly by its type. As a consequence for semigroups with $d=4$ satisfying the condition $(*)$ we have $\mu ({\rm in}(I))\leq 2(t(H))+1$.

A formula for jumping numbers in a two-dimensional regular local ring

In this article we give an explicit formula for the jumping numbers of an ideal of finite colength in a two-dimensional regular local ring with an algebraically closed residue field. For this purpose, we associate a certain numerical semigroup to each vertex of the dual graph of a log-resolution of the ideal.

Factorizations in numerical semigroup algebras

We study a numerical semigroup ring as an algebra over another numerical semigroup ring. The complete intersection property of numerical semigroup algebras is investigated using factorizations of monomials into minimal ones. The goal is to study whether a flat rectangular algebra is a complete intersection. Along this direction, special types of algebras generated by few monomials are worked out in detail.

BETTI NUMBERS FOR CERTAIN COHEN–MACAULAY TANGENT CONES

We compute Betti numbers for a Cohen–Macaulay tangent cone of a monomial curve in the affine $4$-space corresponding to a pseudo-symmetric numerical semigroup. As a byproduct, we also show that for these semigroups, being of homogeneous type and homogeneous are equivalent properties.

Arf good semigroups with fixed conductor

In this paper we present a new algorithm to calculate the set of Arf numerical semigroups with given conductor. Moreover, we extend the previous problem to the class of good semigroups, presenting procedures to compute the set of Arf good semigroups of $$\mathbb {N}^r$$ with a fixed conductor $$\mathbf c \in \mathbb {N}^r$$ , for all $$r \in \mathbb {N}$$ .

On arithmetical numerical monoids with some generators omitted

Numerical monoids (cofinite, additive submonoids of the non-negative integers) arise frequently in additive combinatorics, and have recently been studied in the context of factorization theory. Arithmetical numerical monoids, which are minimally generated by arithmetic sequences, are particularly well-behaved, admitting closed forms for many invariants that are difficult to compute in the general case. In this paper, we answer the question “when does omitting generators from an arithmetical numerical monoid S preserve its (well-understood) set of length sets and/or Frobenius number?” in two extremal cases: (1) we characterize which individual generators can be omitted from S without changing the set of length sets or Frobenius number; and (2) we prove that under certain conditions, nearly every generator of S can be omitted without changing its set of length sets or Frobenius number.

A combinatorial problem and numerical semigroups

Let $a=(a_1,\ldots,a_n)$ and $b=(b_1,\ldots,b_n)$ be two $n$-tuples of positive integers, let $X$ be a set of positive integers, and let $g$ be a positive integer. In this work we show an algorithmic process in order to compute all the sets $C$ of positive integers that fulfill the following conditions: 1) the cardinality of $C$ is equal to $g$; 2) if $x,y\in \mathbb{N} \setminus \{0\}$ and $x+y\in C$, then $C \cap \{x,y\} \neq \emptyset$; 3) if $x \in C$ and $\frac{x-b_i}{a_i} \in \mathbb{N} \setminus \{0\}$ for some $i\in \{1,\ldots,n\}$, then $\frac{x-b_i}{a_i} \in C$; 4) $X \cap C = \emptyset$.

The structure of the minimal free resolution of semigroup rings obtained by gluing

We construct a minimal free resolution of the semigroup ring k[C] in terms of minimal resolutions of k[A] and k[B] when 〈C〉 is a numerical semigroup obtained by gluing two numerical semigroups 〈A〉 and 〈B〉. Using our explicit construction, we compute the Betti numbers, graded Betti numbers, regularity and Hilbert series of k[C], and prove that the minimal free resolution of k[C] has a differential graded algebra structure provided the resolutions of k[A] and k[B] possess them. We discuss the consequences of our results in small embedding dimensions. Finally, we give an extension of our main result to semigroups in Nn.

Frobenius restricted varieties in numerical semigroups

The common behaviour of many families of numerical semigroups led up to defining, firstly, the Frobenius varieties and, secondly, the (Frobenius) pseudo-varieties. However, some interesting families are still out of these definitions. To overcome this situation, here we introduce the concept of restricted Frobenius variety (or R-variety). We will generalize most of the results for varieties and pseudo-varieties to R-varieties. In particular, we will study the tree structure that arise within them.

Wilf’s conjecture and Macaulay’s theorem

Let S \subseteq {\mathbb N} be a numerical semigroup with multiplicity m=\mathrm {min}(S\setminus \{0\}) , conductor c=\mathrm {max}({\mathbb N} \setminus S)+1 and minimally generated by e elements. Let L be the set of elements of S which are smaller than c . Wilf conjectured in 1978 that |L| is bounded below by c/e . We show here that if c \le 3m , then S satisfies Wilf's conjecture. Combined with a recent result of Zhai, this implies that the conjecture is asymptotically true as the genus g(S)=|{\mathbb N} \setminus S| goes to infinity. One main tool in this paper is a classical theorem of Macaulay on the growth of Hilbert functions of standard graded algebras.

Pseudo-Frobenius numbers versus defining ideals in numerical semigroup rings

The structure of the defining ideal of the semigroup ring k[H] of a numerical semigroup H over a field k is described, when the pseudo-Frobenius numbers of H are multiples of a fixed integer.

Homogeneous numerical semigroups

We introduce the concept of homogeneous numerical semigroups and show that all homogeneous numerical semigroups with Cohen-Macaulay tangent cones are of homogeneous type. In embedding dimension three, we classify all numerical semigroups of homogeneous type in to numerical semigroups with complete intersection tangent cones and the homogeneous ones with Cohen-Macaulay tangent cones. We also study the behavior of the homogeneous property by gluing and shiftings to construct a large family of homogeneous numerical semigroups with Cohen-Macaulay tangent cones. In particular we show that these properties fulfill assymptotically in the shifting classes. Several explicit examples are provided along the paper to illustrate the property.

On the second Feng-Rao distance of Algebraic Geometry codes related to Arf semigroups

We describe the second (generalized) Feng-Rao distance for elements in an Arf numerical semigroup that are greater than or equal to the conductor of the semigroup. This provides a lower bound for the second Hamming weight for one point AG codes. In particular, we can obtain the second Feng-Rao distance for the codes defined by asymptotically good towers of function fields whose Weierstrass semigroups are inductive. In addition, we compute the second Feng-Rao number, and provide some examples and comparisons with previous results on this topic. These calculations rely on Ap\'{e}ry sets, and thus several results concerning Ap\'ery sets of Arf semigroups are presented.

Numerical semigroups of Szemerédi type

Given any length k≥3 and density 0<δ≤1, we introduce and study the set Sz(k,δ) consisting of all positive integers n such that every subset of {1,2,…,n} of density at least δ contains an arithmetic progression of length k. A famous theorem of Szemerédi guarantees that this set is not empty. We show that Sz(k,δ)∪{0} is a numerical semigroup and we determine it for (k,δ)=(4,1∕2) and for more than thirty pairs (3,δ) with δ>1∕5.

Characterizations of numerical semigroup complements via Apéry sets

In this paper, we generalize the work of Tuenter to give an identity which completely characterizes the complement of a numerical semigroup in terms of its Apery sets. Using this result, we compute the mth power Sylvester and alternating Sylvester sums for free numerical semigroups. Explicit formulas are given for small m.

Apéry sets of shifted numerical monoids

A numerical monoid is an additive submonoid of the non-negative integers. Given a numerical monoid S, consider the family of “shifted” monoids Mn obtained by adding n to each generator of S. In this paper, we characterize the Apéry set of Mn in terms of the Apéry set of the base monoid S when n is sufficiently large. We give a highly efficient algorithm for computing the Apéry set of Mn in this case, and prove that several numerical monoid invariants, such as the genus and Frobenius number, are eventually quasipolynomial as a function of n.

Near-misses in Wilf’s conjecture

Let S $\subseteq$ N be a numerical semigroup with multiplicity m, conductor c and minimal generating set P. Let L = S $\cap$ [0, c -- 1] and W(S) = |P||L| -- c. In 1978, Herbert Wilf asked whether W(S) $\ge$ 0 always holds, a question known as Wilf's conjecture and open since then. A related number W0(S), satisfying W0(S) $\le$ W(S), has recently been introduced. We say that S is a near-miss in Wilf's conjecture if W0(S) < 0. Near-misses are very rare. Here we construct infinite families of them, with c = 4m and W0(S) arbitrarily small, and we show that the members of these families still satisfy Wilf's conjecture.

Note on a theorem of Eliahou

From a result (Wilf’s conjecture and Macaulay’s theorem, 2017, Theorem 5.11) of Eliahou on the growth of the Hilbert function of a standard graded algebra we derive an inequality related to a question in Wilf (Am Math Mon 85, 1978). This enables us to construct a (to our knowledge new) class of numerical semigroups of embedding dimension 5 and arbitrarily high Cohen-Macaulay-type, for which the question of Wilf (1978) has an affirmative answer.

Dilatations of numerical semigroups

This paper is focused on numerical semigroups and presents a simple construction, that we call “dilatation”, which, from a starting semigroup S, permits to get an infinite family of semigroups which share several properties with S. The invariants of each semigroup T of this family are given in terms of the corresponding invariants of S and both the Apery set and the minimal generators of T are described. We also study three properties that are close to the Gorenstein property of the associated semigroup ring: almost Gorenstein, 2-AGL, and nearly Gorenstein properties. More precisely, we prove that S satisfies one of these properties if and only if each dilatation of S satisfies the corresponding one.

A realization theorem for sets of lengths in numerical monoids

Abstract We show that for every finite nonempty subset of ℕ ≥ 2 {\mathbb{N}_{\geq 2}} there are a numerical monoid H and a squarefree element a ∈ H {a\in H} whose set of lengths 𝖫 ⁢ ( a ) {\mathsf{L}(a)} is equal to L.