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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Arithmetical rank of toric ideals associated to graphs


Author: Anargyros Katsabekis
Journal: Proc. Amer. Math. Soc. 138 (2010), 3111-3123
MSC (2010): Primary 14M25, 13F20, 05C99
Published electronically: May 26, 2010
Previous version: Original version posted May 17, 2010
Corrected version: Current version corrects publisher's introduction of misspelling of "arithmetical" in the abstract.
MathSciNet review: 2653936
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Abstract: Let $ I_{G} \subset K[x_{1},\ldots,x_{m}]$ be the toric ideal associated to a finite graph $ G$. In this paper we study the binomial arithmetical rank and the $ G$-homogeneous arithmetical rank of $ I_G$ in 2 cases:

  1. $ G$ is bipartite,
  2. $ I_G$ is generated by quadratic binomials.
In both cases we prove that the binomial arithmetical rank and the $ G$-homogeneous arithmetical rank coincide with the minimal number of generators of $ I_G$.


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Additional Information

Anargyros Katsabekis
Affiliation: Department of Mathematics, University of the Aegean, 83200 Karlovassi, Samos, Greece
Email: katsabek@aegean.gr

DOI: http://dx.doi.org/10.1090/S0002-9939-2010-10335-0
PII: S 0002-9939(2010)10335-0
Keywords: Arithmetical rank, toric ideals, graphs
Received by editor(s): December 16, 2008
Received by editor(s) in revised form: December 16, 2009
Published electronically: May 26, 2010
Communicated by: Bernd Ulrich
Article copyright: © Copyright 2010 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.