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

 

Minimal systems of binomial generators and the indispensable complex of a toric ideal


Authors: Hara Charalambous, Anargyros Katsabekis and Apostolos Thoma
Journal: Proc. Amer. Math. Soc. 135 (2007), 3443-3451
MSC (2000): Primary 13F20, 05C99
Published electronically: July 3, 2007
MathSciNet review: 2336556
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Abstract: Let $ A=\{{\bf a}_1,\ldots,{\bf a}_m\} \subset \mathbb{Z}^n$ be a vector configuration and $ I_A \subset K[x_1,\ldots,x_m]$ its corresponding toric ideal. The paper consists of two parts. In the first part we completely determine the number of different minimal systems of binomial generators of $ I_A$. In the second part we associate to $ A$ a simplicial complex $ \Delta _{\ind(A)}$. We show that the vertices of $ \Delta _{\ind(A)}$ correspond to the indispensable monomials of the toric ideal $ I_A$, while one dimensional facets of $ \Delta _{\ind(A)}$ with minimal binomial $ A$-degree correspond to the indispensable binomials of $ I_{A}$.


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Additional Information

Hara Charalambous
Affiliation: Department of Mathematics, Aristotle University of Thessaloniki, Thessaloniki 54124, Greece
Email: hara@math.auth.gr

Anargyros Katsabekis
Affiliation: Department of Mathematics, University of Ioannina, Ioannina 45110, Greece
Email: akatsabekis@in.gr

Apostolos Thoma
Affiliation: Department of Mathematics, University of Ioannina, Ioannina 45110, Greece
Email: athoma@cc.uoi.gr

DOI: http://dx.doi.org/10.1090/S0002-9939-07-09037-5
PII: S 0002-9939(07)09037-5
Keywords: Toric ideal, minimal systems of generators, indispensable monomials, indispensable binomials
Received by editor(s): July 10, 2006
Published electronically: July 3, 2007
Additional Notes: This research was co-funded by the European Union in the framework of the program “Pythagoras" of the “Operational Program for Education and Initial Vocational Training" of the 3rd Community Support Framework of the Hellenic Ministry of Education.
Communicated by: Bernd Ulrich
Article copyright: © Copyright 2007 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.