scholarly journals Cyclotomic numerical semigroup polynomials with at most two irreducible factors

2021 ◽  
Author(s):  
Alessio Borzì ◽  
Andrés Herrera-Poyatos ◽  
Pieter Moree
Keyword(s):  
Complete Intersection ◽  
Numerical Semigroup ◽  
Discrete Math ◽  
Cyclotomic Polynomials ◽  
Embedding Dimension

AbstractA numerical semigroup S is cyclotomic if its semigroup polynomial $$\mathrm {P}_S$$ P S is a product of cyclotomic polynomials. The number of irreducible factors of $$\mathrm {P}_S$$ P S (with multiplicity) is the polynomial length $$\ell (S)$$ ℓ ( S ) of S. We show that a cyclotomic numerical semigroup is complete intersection if $$\ell (S)\le 2$$ ℓ ( S ) ≤ 2 . This establishes a particular case of a conjecture of Ciolan et al. (SIAM J Discrete Math 30(2):650–668, 2016) claiming that every cyclotomic numerical semigroup is complete intersection. In addition, we investigate the relation between $$\ell (S)$$ ℓ ( S ) and the embedding dimension of S.

On the numerical semigroup generated by {bn+1+i+bn+i−1b−1∣i∈N}$\begin{array}{} \{b^{n+1+i}+\frac{b^{n+i}-1}{b-1}\mid i\in\mathbb{N}\} \end{array}$

2020 ◽  
Vol 30 (4) ◽  
pp. 257-264
Author(s):  
Ze Gu
Keyword(s):  
Numerical Semigroup ◽  
Frobenius Number ◽  
Embedding Dimension ◽  
Order Relations ◽  
Positive Integers

AbstractLet b, n be two positive integers such that b ≥ 2, and S(b, n) be the numerical semigroup generated by $\begin{array}{} \{b^{n+1+i}+\frac{b^{n+i}-1}{b-1}\mid i\in\mathbb{N}\} \end{array}$. Applying two order relations, we give formulas for computing the embedding dimension, the Frobenius number, the type and the genus of S(b, n).

scholarly journals Proof of de Smit’s conjecture: a freeness criterion

2017 ◽  
Vol 153 (11) ◽  
pp. 2310-2317
Author(s):  
Sylvain Brochard
Keyword(s):  
Complete Intersection ◽  
Local Rings ◽  
Embedding Dimension ◽  
Fermat's Last Theorem ◽  
Multiplicity One ◽  
Fermat’S Last Theorem ◽  
Invent Math

Let $A\rightarrow B$ be a morphism of Artin local rings with the same embedding dimension. We prove that any $A$-flat $B$-module is $B$-flat. This freeness criterion was conjectured by de Smit in 1997 and improves Diamond’s criterion [The Taylor–Wiles construction and multiplicity one, Invent. Math. 128 (1997), 379–391, Theorem 2.1]. We also prove that if there is a nonzero $A$-flat $B$-module, then $A\rightarrow B$ is flat and is a relative complete intersection. Then we explain how this result allows one to simplify Wiles’s proof of Fermat’s last theorem: we do not need the so-called ‘Taylor–Wiles systems’ any more.

scholarly journals Measuring primality in numerical semigroups with embedding dimension three

2015 ◽  
Vol 15 (01) ◽  
pp. 1650007 ◽  
Author(s):  
S. T. Chapman ◽  
P. A. García-Sánchez ◽  
Z. Tripp ◽  
C. Viola
Keyword(s):  
Numerical Semigroup ◽  
Numerical Semigroups ◽  
Embedding Dimension ◽  
Catenary Degree

In this paper, we find the ω-value of the generators of any numerical semigroup with embedding dimension three. This allows us to determine all possible orderings of the ω-values of the generators. In addition, we relate the ω-value of the numerical semigroup to its catenary degree.

Delta sets for nonsymmetric numerical semigroups with embedding dimension three

2018 ◽  
Vol 30 (1) ◽  
pp. 15-30 ◽  
Author(s):  
Pedro A. García-Sánchez ◽  
David Llena ◽  
Alessio Moscariello
Keyword(s):  
Fast Algorithm ◽  
Numerical Semigroup ◽  
Numerical Semigroups ◽  
Embedding Dimension ◽  
Delta Set

Abstract We present a fast algorithm to compute the Delta set of a nonsymmetric numerical semigroup with embedding dimension three. We also characterize the sets of integers that are the Delta set of a numerical semigroup of this kind.

scholarly journals Factoring in Embedding Dimension Three Numerical Semigroups

10.37236/410 ◽  
2010 ◽  
Vol 17 (1) ◽  
Author(s):  
F. Aguiló-Gost ◽  
P. A. García-Sánchez
Keyword(s):  
Numerical Semigroup ◽  
Main Tool ◽  
Numerical Semigroups ◽  
Embedding Dimension ◽  
Catenary Degree

Let us consider a $3$-numerical semigroup $S=\langle{a,b,N}\rangle$. Given $m\in S$, the triple $(x,y,z)\in\mathbb{N}^3$ is a factorization of $m$ in $S$ if $xa+yb+zN=m$. This work is focused on finding the full set of factorizations of any $m\in S$ and as an application we compute the catenary degree of $S$. To this end, we relate a 2D tessellation to $S$ and we use it as a main tool.

Elements in a Numerical Semigroup with Factorizations of the Same Length

2011 ◽  
Vol 54 (1) ◽  
pp. 39-43 ◽  
Author(s):  
S. T. Chapman ◽  
P. A. García-Sánchez ◽  
D. Llena ◽  
J. Marshall
Keyword(s):  
Recent Literature ◽  
Numerical Semigroup ◽  
Embedding Dimension ◽  
Numerical Monoid ◽  
Irreducible Elements

AbstractQuestions concerning the lengths of factorizations into irreducible elements in numerical monoids have gained much attention in the recent literature. In this note, we show that a numerical monoid has an element with two different irreducible factorizations of the same length if and only if its embedding dimension is greater than two. We find formulas in embedding dimension three for the smallest element with two different irreducible factorizations of the same length and the largest element whose different irreducible factorizations all have distinct lengths. We show that these formulas do not naturally extend to higher embedding dimensions.

scholarly journals Presentations of a numerical semigroup

2020 ◽  
Vol 8 (1) ◽  
pp. 1
Author(s):  
Belgin Özer ◽  
Sibel Kanbay
Keyword(s):  
Complete Intersection ◽  
Numerical Semigroup ◽  
Numerical Semigroups ◽  
Catenary Degree ◽  
Minimal Presentations

In this paper, we mainly study the minimal presentations of numerical semigroups. Moreover, we examine the concept of gluing, complete intersection, catenary degree, elasticity of some numerical semigroups.  

Sign in / Sign up

Export Citation Format

Share Document