Numerical semigroups: Apéry sets and Hilbert series

2009 ◽  
Vol 79 (2) ◽  
pp. 323-340 ◽  
Author(s):  
Jorge L. Ramírez Alfonsín ◽  
Øystein J. Rødseth
Keyword(s):  
Hilbert Series ◽  
Numerical Semigroups ◽  
Apéry Sets

scholarly journals Apery Sets of Numerical Semigroups#

2005 ◽  
Vol 33 (10) ◽  
pp. 3831-3838 ◽  
Author(s):  
Monica Madero-Craven ◽  
Kurt Herzinger
Keyword(s):  
Numerical Semigroups ◽  
Apéry Sets

scholarly journals THE SHORT RESOLUTION OF A SEMIGROUP ALGEBRA

2017 ◽  
Vol 96 (3) ◽  
pp. 400-411 ◽  
Author(s):  
I. OJEDA ◽  
A. VIGNERON-TENORIO
Keyword(s):  
Semigroup Algebra ◽  
Numerical Semigroups ◽  
Frobenius Numbers ◽  
Affine Semigroups ◽  
Affine Semigroup ◽  
Lattice Ideal ◽  
Apéry Sets

This work generalises the short resolution given by Pisón Casares [‘The short resolution of a lattice ideal’, Proc. Amer. Math. Soc.131(4) (2003), 1081–1091] to any affine semigroup. We give a characterisation of Apéry sets which provides a simple way to compute Apéry sets of affine semigroups and Frobenius numbers of numerical semigroups. We also exhibit a new characterisation of the Cohen–Macaulay property for simplicial affine semigroups.

Short Generating Functions for some Semigroup Algebras

10.37236/1729 ◽  
2003 ◽  
Vol 10 (1) ◽  
Author(s):  
Graham Denham
Keyword(s):  
Rational Function ◽  
Generating Functions ◽  
Hilbert Series ◽  
Simple Expression ◽  
Arbitrary Field ◽  
Graded Algebra ◽  
Semigroup Algebras ◽  
Positive Integers

Let $a_1,\ldots,a_n$ be distinct, positive integers with $(a_1,\ldots,a_n)=1$, and let k be an arbitrary field. Let $H(a_1,\ldots,a_n;z)$ denote the Hilbert series of the graded algebra k$[t^{a_1},t^{a_2},\ldots,t^{a_n}]$. We show that, when $n=3$, this rational function has a simple expression in terms of $a_1,a_2,a_3$; in particular, the numerator has at most six terms. By way of contrast, it is known that no such expression exists for any $n\geq4$.

scholarly journals The Hilbert Series of a Linear Symplectic Circle Quotient

2014 ◽  
Vol 23 (1) ◽  
pp. 46-65 ◽  
Author(s):  
Hans-Christian Herbig ◽  
Christopher Seaton
Keyword(s):  
Hilbert Series

scholarly journals 2, 12, 117, 1959, 45171, 1170086, …: a Hilbert series for the QCD chiral Lagrangian

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Lukáš Gráf ◽  
Brian Henning ◽  
Xiaochuan Lu ◽  
Tom Melia ◽  
Hitoshi Murayama
Keyword(s):  
Hilbert Series ◽  
Charge Conjugation ◽  
External Fields ◽  
Dynkin Diagrams ◽  
Non Linear ◽  
Very High ◽  
Chiral Lagrangian

Abstract We apply Hilbert series techniques to the enumeration of operators in the mesonic QCD chiral Lagrangian. Existing Hilbert series technologies for non-linear realizations are extended to incorporate the external fields. The action of charge conjugation is addressed by folding the $$ \mathfrak{su}(n) $$ su n Dynkin diagrams, which we detail in an appendix that can be read separately as it has potential broader applications. New results include the enumeration of anomalous operators appearing in the chiral Lagrangian at order p8, as well as enumeration of CP-even, CP-odd, C-odd, and P-odd terms beginning from order p6. The method is extendable to very high orders, and we present results up to order p16.(The title sequence is the number of independent C-even and P-even operators in the mesonic QCD chiral Lagrangian with three light flavors of quarks, at chiral dimensions p2, p4, p6, …)

scholarly journals Almost canonical ideals and GAS numerical semigroups

2021 ◽  
pp. 1-24
Author(s):  
Marco D’Anna ◽  
Francesco Strazzanti
Keyword(s):  
Numerical Semigroups
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