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), 34433451
MSC (2000):
Primary 13F20, 05C99
Published electronically:
July 3, 2007
MathSciNet review:
2336556
Fulltext PDF Free Access
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Abstract: Let be a vector configuration and 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 . In the second part we associate to a simplicial complex . We show that the vertices of correspond to the indispensable monomials of the toric ideal , while one dimensional facets of with minimal binomial degree correspond to the indispensable binomials of .
 1.
S. Aoki, A. Takemura and R. Yoshida, Indispensable monomials of toric ideals and Markov bases, preprint 2005.
 2.
Imre
Bárány and Herbert
Scarf, Matrices with identical sets of neighbors, Math. Oper.
Res. 23 (1998), no. 4, 863–873. MR 1662422
(2000b:90030), http://dx.doi.org/10.1287/moor.23.4.863
 3.
E.
Briales, A.
Campillo, C.
Marijuán, and P.
Pisón, Minimal systems of generators for ideals of
semigroups, J. Pure Appl. Algebra 124 (1998),
no. 13, 7–30. MR 1600261
(98k:20105), http://dx.doi.org/10.1016/S00224049(96)001065
 4.
Antonio
Campillo and Pilar
Pisón, L’idéal d’un semigroupe de type
fini, C. R. Acad. Sci. Paris Sér. I Math. 316
(1993), no. 12, 1303–1306 (French, with English and French
summaries). MR
1226120 (94b:20055)
 5.
A. Cayley, A theorem on trees, Quart. J. Math. 23 (1889) 376378.
 6.
CoCoATeam, CoCoA: a system for doing Computations in Commutative Algebra, available at http://cocoa.dima.unige.it.
 7.
Persi
Diaconis and Bernd
Sturmfels, Algebraic algorithms for sampling from conditional
distributions, Ann. Statist. 26 (1998), no. 1,
363–397. MR 1608156
(99j:62137), http://dx.doi.org/10.1214/aos/1030563990
 8.
J.
H. van Lint and R.
M. Wilson, A course in combinatorics, Cambridge University
Press, Cambridge, 1992. MR 1207813
(94g:05003)
 9.
Ezra
Miller and Bernd
Sturmfels, Combinatorial commutative algebra, Graduate Texts
in Mathematics, vol. 227, SpringerVerlag, New York, 2005. MR 2110098
(2006d:13001)
 10.
Hidefumi
Ohsugi and Takayuki
Hibi, Indispensable binomials of finite graphs, J. Algebra
Appl. 4 (2005), no. 4, 421–434. MR 2166253
(2006e:13023), http://dx.doi.org/10.1142/S0219498805001265
 11.
H. Ohsugi and T. Hibi, Toric ideals arising from contingency tables, Proceedings of the Ramanujan Mathematical Society's Lecture Notes Series, 2006, pp. 87111.
 12.
Irena
Peeva and Bernd
Sturmfels, Generic lattice ideals, J. Amer. Math. Soc. 11 (1998), no. 2, 363–373. MR 1475887
(98i:13022), http://dx.doi.org/10.1090/S0894034798002550
 13.
B. Sturmfels, Gröbner Bases and Convex Polytopes, University Lecture Series, No. 8 American Mathematical Society, Providence, R.I., 1995.
 14.
Akimichi
Takemura and Satoshi
Aoki, Some characterizations of minimal Markov basis for sampling
from discrete conditional distributions, Ann. Inst. Statist. Math.
56 (2004), no. 1, 1–17. MR 2053726
(2005g:62103), http://dx.doi.org/10.1007/BF02530522
 1.
 S. Aoki, A. Takemura and R. Yoshida, Indispensable monomials of toric ideals and Markov bases, preprint 2005.
 2.
 I. Barany and H. Scarf, Matrices with identical sets of neighbors, Mathematics of Operation Research 23 (1998) 863873. MR 1662422 (2000b:90030)
 3.
 E. Briales, A. Campillo, C. Marijuán and P. Pisón, Minimal systems of generators for ideals of semigroups, J. Pure Appl. Algebra 124 (1998) 730. MR 1600261 (98k:20105)
 4.
 A. Campillo and P. Pison, L'idéal d'un semigroup de type fini, Comptes Rendues Acad. Sci. Paris, Série I, 316 (1993) 13031306. MR 1226120 (94b:20055)
 5.
 A. Cayley, A theorem on trees, Quart. J. Math. 23 (1889) 376378.
 6.
 CoCoATeam, CoCoA: a system for doing Computations in Commutative Algebra, available at http://cocoa.dima.unige.it.
 7.
 P. Diaconis and B. Sturmfels, Algebraic algorithms for sampling from conditional distributions, Ann. Statist., 26 (1) (1998) 363397. MR 1608156 (99j:62137)
 8.
 J.H. van Lint and R.M Wilson, A Course in Combinatorics, Cambridge University Press, 1992. MR 1207813 (94g:05003)
 9.
 E. Miller and B. Sturmfels, Combinatorial Commutative Algebra, Graduate Texts in Mathematics, vol. 227, SpringerVerlag, New York, 2005. MR 2110098 (2006d:13001)
 10.
 H. Ohsugi and T. Hibi, Indispensable binomials of finite graphs, J. Algebra Appl. 4 (2005), no. 4, 421434. MR 2166253 (2006e:13023)
 11.
 H. Ohsugi and T. Hibi, Toric ideals arising from contingency tables, Proceedings of the Ramanujan Mathematical Society's Lecture Notes Series, 2006, pp. 87111.
 12.
 I. Peeva and B. Sturmfels, Generic Lattice Ideals, J. Amer. Math. Soc. 11 (1998) 363373. MR 1475887 (98i:13022)
 13.
 B. Sturmfels, Gröbner Bases and Convex Polytopes, University Lecture Series, No. 8 American Mathematical Society, Providence, R.I., 1995.
 14.
 A. Takemura and S. Aoki, Some characterizations of minimal Markov basis for sampling from discrete conditional distributions, Ann. Inst. Statist. Math., 56 (1)(2004) 117. MR 2053726 (2005g:62103)
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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/S0002993907090375
PII:
S 00029939(07)090375
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 cofunded 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.
