Symmetric semigroups of integers generated by 4 elements

Bresinsky, H.

doi:10.1007/bf01170309

Cited by 97 publications

(95 citation statements)

References 2 publications

Supporting

Mentioning

Contrasting

Unclassified

Order By: Relevance

“…In this section, we first recall Bresinsky's theorem, which gives the explicit description of the defining ideal of a Gorenstein monomial curve with embedding dimension four in the non-complete intersection case [3,Theorem 3]. Theorem 2.1 (Bresinsky's Theorem).…”

Section: Non-complete Intersection Casementioning

confidence: 99%

“…Thus, in this case, we also say that C is a Gorenstein monomial curve with embedding dimension four. In 1975, Bresinsky not only showed that the ideal I(C) of the curve C is minimally generated by either 3 (complete intersection case) or 5 (non-complete intersection case) elements, but also gave an explicit description of the defining ideal I(C), see [3]. Knowing the defining ideal I(C) explicitly by the work of Bresinsky, we investigate the generators of the ideal I(C) * , which is generated by the polynomials f * for f in I(C), where f * is the homogeneous summand of f of least degree.…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Hilbert functions of Gorenstein monomial curves

Arslan

Mete

2007

Proc. Amer. Math. Soc.

View full text Add to dashboard Cite

Abstract. It is a conjecture due to M. E. Rossi that the Hilbert function of a one-dimensional Gorenstein local ring is non-decreasing. In this article, we show that the Hilbert function is non-decreasing for local Gorenstein rings with embedding dimension four associated to monomial curves, under some arithmetic assumptions on the generators of their defining ideals in the noncomplete intersection case. In order to obtain this result, we determine the generators of their tangent cones explicitly by using standard basis computations under these arithmetic assumptions and show that the tangent cones are Cohen-Macaulay. In the complete intersection case, by characterizing certain families of complete intersection numerical semigroups, we give an inductive method to obtain large families of complete intersection local rings with arbitrary embedding dimension having non-decreasing Hilbert functions.

show abstract

Section: Non-complete Intersection Casementioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Hilbert functions of Gorenstein monomial curves

Arslan

Mete

2007

Proc. Amer. Math. Soc.

View full text Add to dashboard Cite

show abstract

“…This proves that the cardinality of a minimal presentation for a numerical semigroup cannot be bounded as a function of its embedding dimension. Bresinsky also proves in [3] that the cardinality of a minimal presentation for a symmetric numerical semigroup with embedding dimension 4 can only be 3 or 5. This makes us speculate about the possibility that the cardinality of a minimal presentation for a symmetric numerical semigroup can be bounded in a function of its embedding dimension and a candidate for it is…”

Section: Introductionmentioning

confidence: 99%

Symmetric numerical semigroups with arbitrary multiplicity and embedding dimension

Rosales

2001

Proc. Amer. Math. Soc.

View full text Add to dashboard Cite

show abstract

“…Another characterization of symmetric numerical semigroups is the following (see [3]). The numerical semigroup S is symmetric if the greatest element, w, of S(a) satisfies the condition that for every s ∈ S(a), the element w − s is in S. We use this result to give a characterization of the numerical semigroups generated by intervals that are symmetric.…”

Section: ) Complete Intersection Semigroupsmentioning

confidence: 99%