Two way subtable sum problems and quadratic Gröbner bases

Authors:
Hidefumi Ohsugi and Takayuki Hibi

Journal:
Proc. Amer. Math. Soc. **137** (2009), 1539-1542

MSC (2000):
Primary 13P10

DOI:
https://doi.org/10.1090/S0002-9939-08-09675-5

Published electronically:
December 5, 2008

MathSciNet review:
2470810

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Abstract | References | Similar Articles | Additional Information

Abstract: Hara, Takemura and Yoshida discussed toric ideals arising from two way subtable sum problems and showed that these toric ideals are generated by quadratic binomials if and only if the subtables are either diagonal or triangular. In the present paper, we show that if the subtables are either diagonal or triangular, then their toric ideals possess quadratic Gröbner bases.

**1.**David Cox, John Little, and Donal O’Shea,*Ideals, varieties, and algorithms*, Undergraduate Texts in Mathematics, Springer-Verlag, New York, 1992. An introduction to computational algebraic geometry and commutative algebra. MR**1189133****2.**H. Hara, A. Takemura and R. Yoshida, Markov bases for two-way subtable sum problems, arXiv:math.CO/0708.2312v1, 2007.**3.**Hidefumi Ohsugi and Takayuki Hibi,*Toric ideals generated by quadratic binomials*, J. Algebra**218**(1999), no. 2, 509–527. MR**1705794**, https://doi.org/10.1006/jabr.1999.7918**4.**H. Ohsugi and T. Hibi, Toric ideals arising from contingency tables, in*Commutative Algebra and Combinatorics*, Ramanujan Mathematical Society Lecture Notes Series, Vol. 4, Ramanujan Mathematical Society, Mysore, India, 2007, pp. 91-115.**5.**Bernd Sturmfels,*Gröbner bases and convex polytopes*, University Lecture Series, vol. 8, American Mathematical Society, Providence, RI, 1996. MR**1363949**

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Additional Information

**Hidefumi Ohsugi**

Affiliation:
Department of Mathematics, College of Science, Rikkyo University, Toshima, Tokyo 171-8501, Japan

Email:
ohsugi@rkmath.rikkyo.ac.jp

**Takayuki Hibi**

Affiliation:
Department of Pure and Applied Mathematics, Graduate School of Information Science and Technology, Osaka University, Toyonaka, Osaka 560-0043, Japan

Email:
hibi@math.sci.osaka-u.ac.jp

DOI:
https://doi.org/10.1090/S0002-9939-08-09675-5

Keywords:
Quadratic Gr\"obner bases,
toric ideals

Received by editor(s):
December 2, 2007

Received by editor(s) in revised form:
April 29, 2008, and June 13, 2008

Published electronically:
December 5, 2008

Communicated by:
Bernd Ulrich

Article copyright:
© Copyright 2008
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication.