Numerical semigroups of maximal and almost maximal length

Brown, W. C.; Curtis, Frank

doi:10.1007/bf02573421

Cited by 8 publications

(7 citation statements)

References 3 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…As for higher dimension, m ≥ 4, relations between G(d m ) and G(d m ) exist [4,5] and are similar to those given in Theorems 1 and 2,…”

Section: Consider the 3-dim Version Of Conjecture 1 And Recall Recentsupporting

confidence: 72%

See 2 more Smart Citations

Weak asymptotics in the 3-dim Frobenius problem

Fel

2008

Funct. Anal. Other Math.

View full text Add to dashboard Cite

We consider the Frobenius problem for the 3-dim numerical semigroup S(d 1 , d 2 , d 3 ) generated by three positive integers d 1 , d 2 , d 3 with gcd(d 1 , d 2 , d 3 ) = 1 and calculate weak (Cesaro) asymptotics of its conductor C(d 1 , d 2 , d 3 ) and the fraction p(d 1 , d 2 , d 3 ) of the segment [0; C(d 1 , d 2 , d 3 ) − 1] occupied by the semigroup. Four conjectures # 1999-8, # 1999-9, # 1999-10 and # 2003-5 which were posed by V.I. Arnold (Arnold's Problems, pp. 129-130, 163, 2004) and devoted to the statistics of the m-dim semigroups, m ≥ 3, are refuted for the case m = 3.

show abstract

“…As for higher dimension, m ≥ 4, relations between G(d m ) and G(d m ) exist [4,5] and are similar to those given in Theorems 1 and 2,…”

Section: Consider the 3-dim Version Of Conjecture 1 And Recall Recentsupporting

confidence: 72%

“…The set of remaining triples d 3 gives rise to nonsymmetric semigroups which satisfy the relation [5] …”

Section: Algebra Of Numerical Semigroups S(d 1 D D 3 )mentioning

confidence: 99%

See 1 more Smart Citation

Weak asymptotics in the 3-dim Frobenius problem

Fel

2008

Funct. Anal. Other Math.

View full text Add to dashboard Cite

show abstract

“…[5] or [6]) thus, rings of maximal embedding dimension and with type at least two have rational P A Der k A (z). If k[S] is of almost maximal length, then S = 4, 5, 11 , S = 4, 7, 13 , S = 3, 3d + 2, 3d + 4 for some d 1, or S = p, dp + 1, dp + 2, .…”

Section: The Case Of Maximal Embedding Dimension Maximal Length or mentioning

confidence: 96%

The Poincaré series of the module of derivations of affine monomial curves

Fröberg

Micale

2004

Journal of Algebra

View full text Add to dashboard Cite

Let A be a graded k-algebra and M be a finitely generated A-module. The Poincaré series P A M (z) is the formal power series i dim k Tor A i (k, M)z i . We study the Poincaré series of Der k k[S], the module of derivations of a numerical semigroup ring k[S], and we relate it to the Poincaré series of k over k [S] and to the type of S. We then use this in order to determine the Poincaré series of Der k k [S] or, at least, its rationality, for some classes of examples. We finally give an example of a non-rational P k [S] Der k k[S] (z).

show abstract

“…, mk + m − 1} , k ≥ 1. When #∆ H (d m ) gets an intermediate value in the interval (0, #∆ G (d m ) · [t (d m ) − 1]), semigroup S (d m )can be generated by sophisticated series of generators, similar to (5.22), or by a significant number of sporadic tuples[6]. In the meantime, the study of numerical semigroups with generic length are far from its completeness.Call S (d m ) semigroup of almost minimal length (aml) or almost maximal length (AML) if it has, respectively,(5.23) #∆ H (d m aml ) = 1, or #∆ H (d m AML ) = #∆ G (d m AML ) · [t(d m AML ) − 1] − 1.The semigroups of almost minimal length were discussed in Section 5.1.…”

mentioning

confidence: 98%

Gaps in nonsymmetric numerical semigroups

Aicardi¹,

Fel

2010

Isr. J. Math.

View full text Add to dashboard Cite

There exist two different sorts of gaps in the nonsymmetric numerical additive semigroups finitely generated by a minimal set of positive integers {d 1 , . . . , dm}. The h-gaps are specific only for the nonsymmetric semigroups while the g-gaps are possessed by both, symmetric and nonsymmetric semigroups. We derive the generating functions for the corresponding sets of gaps, ∆ H (d m ) and ∆ G (d m ), and prove several statements on the minimal and maximal values of the h-gaps. Detailed description of both sorts of gaps is given for three special kinds of nonsymmetric semigroups: semigroups with maximal embedding dimension, semigroups of maximal and almost maximal length, and pseudo-symmetric semigroups.

show abstract

Numerical semigroups of maximal and almost maximal length

Cited by 8 publications

References 3 publications

Weak asymptotics in the 3-dim Frobenius problem

Weak asymptotics in the 3-dim Frobenius problem

The Poincaré series of the module of derivations of affine monomial curves

Gaps in nonsymmetric numerical semigroups

Contact Info

Product

Resources

About