Arf numerical semigroups with low multiplicity via Gröbner basis

Volume 32, Issue 2, pp 175--187 http://dx.doi.org/10.22436/jmcs.032.02.07

Publication Date: August 29, 2023 Submission Date: May 29, 2023 Revision Date: July 02, 2023 Accteptance Date: July 11, 2023

Download PDF

Download XML

• Export citations
  • CITATIONS & REFERENCES
  • EndNote (.ENW)
  • Mendeley, JabRef (.BIB)
  • Papers, Zotero, Reference Manager, RefWorks (.RIS)
  • ARTICLE CITATION
  • EndNote (.ENW)
  • Mendeley, JabRef (.BIB)
  • Papers, Zotero, Reference Manager, RefWorks (.RIS)
  • REFERENCES
  • EndNote (.ENW)
  • Mendeley, JabRef (.BIB)
  • Papers, Zotero, Reference Manager, RefWorks (.RIS)

• 355 Downloads
• 1008 Views

Authors

B. Ozer - Department of Mathematics, Gaziantep University, 27310, Gaziantep, Turkey. G. Bahar - Department of Mathematics, Gaziantep University, 27310, Gaziantep, Turkey. M. Albaity - Department of Mathematics, King Abdulaziz University, 21589, Jeddah, Kingdom of Saudi Arabia.

Abstract

In this paper, the Gröbner basis over RF-matrices of Arf numerical semigroups are presented. The Arf properties ideals for the RF-matrices obtained by RF-Relations are provided and the aforementioned concepts are associated through the Gröbner basis of the Arf numerical semigroup. Moreover, we prove that if we have a minimal presentation (or a Gröbner basis of the ideal associated to the semigroup), then this will be a system of generators of the subgroup of \({\mathbb{Z}}^p\) with the equation \(n_1x_1+n_2x_2+\cdots+n_px_p=0\), where \(\{n_1, n_2, \ldots,n_p\}\) are the generators of numerical semigroup \(G\).

Share and Cite

Keywords

• Arf numerical semigroups
• multiplicity
• presentation
• RF (ow factoritazion)-matrices
• Gröbner basis

MSC

•  20M12
•  20M14
•  13P10
•  05A17

References

• [1] W. W. Adams, P. Loustaunau, An Introduction to Gr¨obner Bases, American Mathematical Society, Providence, RI (1994)
  • View Article
  • MathSciNet
  • Google Scholar


• [2] C. Arf, Une interpr´etation alg´ebrique de la suite des ordres de multiplicit´e d’une branche alg´ebrique, Proc. London Math. Soc. (2), 50 (1948), 256–287
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [3] A. Assi, P. A. Garc´ıa-S´anchez, Numerical Semigroups and Applications, Springer, [Cham] (2016)
  • View Article
  • MathSciNet
  • Google Scholar


• [4] V. Barucci, D. E. Dobbs, M. Fontana, Maximality Properties in Numerical Semigroups and Applications to One- Dimensional Analytically Irreducible Local Domains, Mem. Amer. Math. Soc., (1997)
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [5] B. Buchberger, An Algorithmical Criteria for the Solvability of Algebraic Systems of Equations, Aequationes Math., 4 (1970), 374–383
  • View Article
  • MathSciNet
  • Google Scholar


• [6] A. Brauer, On a problem of partitions, Amer. J. Math., 64 (1942), 299-312
  • View Article
  • MathSciNet
  • Google Scholar


• [7] A. C¸ alıs¸kan, Almost Gorenstein Single Curves, Master’s thesis, Balıkesir Institute of Science, Balıkesir, T¨ urkiye, (2020),
  • View Article
  • Google Scholar


• [8] A. Campillo, J. I. Farr´an, C. Munuera, On the parameters of algebraic geometry codes related to Arf semigroups, IEEE Trans. Inform. Theory, 46 (2000), 2634–2638
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [9] P. Du Val, The Jacobian algorithm and the multiplicity sequence of an algebraic branch, Rev. Fac. Sci. Univ. Istanbul (A), 7 (1942), 107–112
  • View Article
  • Google Scholar
  • MATH


• [10] R. Fr¨oberg, C. Gottlieb, R. H¨aggkvist, On numerical semigroups, Semigroup Forum, 35 (1987), 63–83
  • View Article
  • MathSciNet
  • Google Scholar


• [11] , The GAP Group, GAP — Groups, Algorithms, and Programming, Version 4.12.2, , (2022),
  • View Article


• [12] P. A. Garc´ıa-S´anchez, B. A. Heredia, H. ˙I. Karakas¸, J. C. Rosales, Parametrizing Arf Numerical Semigroups, J. Algebra Appl., 16 (2017), 31 pages
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [13] J. Herzog, K. Watanabe, Almost Symmetric Numerical Semigroups, Semigroup Forum, 98 (2019), 589–630
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [14] J. Lipman, Stable ideals and Arf rings, Amer. J. Math., 93 (1971), 649–685
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [15] J. C. Rosales, On numerical semigroups, Semigroup Forum, 52 (1996), 307–318
  • View Article
  • MathSciNet


• [16] J. C. Rosales, M. B. Branco, The Frobenius Problem for Numerical Semigroups with Multiplicity Four, Semigroup Forum, 83 (2011), 468–478
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [17] J. C. Rosales, P. A. Garc´ıa-S´anchez, J. I. Garc´ıa-Garc´ıa, M. B. Branco, Arf Numerical Semigroups, J. Algebra, 276 (2004), 3–12
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [18] J. C. Rosales, P. A. Garc´ıa-S´anchez, Numerical Semigroups, Springer, New York (2009)
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [19] M. S ¨ uer, M. Yes¸il, Row-Factorization Matrices in Arf Numerical Semigroups and Defining Ideals, arXiv:2210.14856v1 [math.AC], (2022), 1–24
  • View Article
  • Google Scholar


• [20] R. H. Villarreal, Monomial Algebras, Marcel Dekker, Inc., New York (2001)
  • View Article
  • Google Scholar