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Complete intersections in toric ideals
Author(s):
Eduardo
Cattani;
Raymond
Curran;
Alicia
Dickenstein
Journal:
Proc. Amer. Math. Soc.
135
(2007),
329-335.
MSC (2000):
Primary 14M10;
Secondary 14M25, 13C40
Posted:
August 1, 2006
MathSciNet review:
2255278
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Abstract:
We present examples that show that in dimension higher than one or codimension higher than two, there exist toric ideals  such that no binomial ideal contained in  and of the same dimension is a complete intersection. This result has important implications in sparse elimination theory and in the study of the Horn system of partial differential equations.
References:
-
- 1.
- M. Barile, M. Morales, and A. Thoma.
On simplicial toric varieties which are set-theoretic complete intersections.
J. Algebra, 226(2):880-892, 2000. MR 1752767 (2001i:14066) - 2.
- M. Barile, M. Morales, and A. Thoma.
On free complete intersections.
In Geometric and combinatorial aspects of commutative algebra (Messina, 1999), volume 217 of Lecture Notes in Pure and Appl. Math., pages 1-9. Dekker, New York, 2001. MR 1824213 (2002c:13026) - 3.
- M. Barile, M. Morales, and A. Thoma.
Set-theoretic complete intersections on binomials.
Proc. Amer. Math. Soc., 130(7):1893-1903, 2002. MR 1896020 (2003f:14058) - 4.
- CoCoATeam.
CoCoA: a system for doing Computations in Commutative Algebra.
Available at http://cocoa.dima.unige.it. - 5.
- C. Delorme.
Sous-monoïdes d'intersection complète de  .
Ann. Sci. École Norm. Sup. (4), 9(1):145-154, 1976. MR 0407038 (53:10821) - 6.
- A. Dickenstein, L. F. Matusevich, and T. Sadykov.
Bivariate hypergeometric  -modules.
Advances in Mathematics, 196(1):78-123, 2005. MR 2159296 - 7.
- A. Dickenstein and B. Sturmfels.
Elimination theory in codimension 2.
J. Symbolic Comput., 34:119-135, 2002. MR 1930829 (2003h:14073) - 8.
- D. Eisenbud and B. Sturmfels.
Binomial ideals.
Duke Math. J., 84(1):1-45, 1996. MR 1394747 (97d:13031) - 9.
- K. G. Fischer, W. Morris, and J. Shapiro.
Affine semigroup rings that are complete intersections.
Proc. Amer. Math. Soc., 125(11):3137-3145, 1997. MR 1401741 (97m:13026) - 10.
- K. G. Fischer, W. Morris, and J. Shapiro.
Mixed dominating matrices.
Linear Algebra Appl., 270:191-214, 1998. MR 1484081 (98j:15029) - 11.
- K. G. Fischer and J. Shapiro.
Mixed matrices and binomial ideals.
J. Pure Appl. Algebra, 113(1):39-54, 1996. MR 1411645 (97h:13008) - 12.
- I.M. Gel'fand, M.M. Kapranov, and A.V. Zelevinsky.
Discriminants, Resultants and Multidimensional Determinants.
Birkhäuser, Boston, 1994. MR 1264417 (95e:14045) - 13.
- J. Herzog.
Generators and relations of abelian semigroups and semigroup rings.
Manuscripta Math., 3:175-193, 1970. MR 0269762 (42:4657) - 14.
- S. Hosten and J. Shapiro.
Primary decomposition of lattice basis ideals.
J. Symbolic Comput., 29(4-5):625-639, 2000.
Symbolic computation in algebra, analysis, and geometry (Berkeley, CA, 1998). MR 1769658 (2001h:13012) - 15.
- H. Nakajima.
Affine torus embeddings which are complete intersections.
Tohoku Math. J. (2), 38(1):85-98, 1986. MR 0826766 (87c:14058) - 16.
- J. C. Rosales and P. A. García-Sánchez.
On complete intersection affine semigroups.
Comm. Algebra, 23(14):5395-5412, 1995. MR 1363611 (96m:14068) - 17.
- G. Scheja, O. Scheja, and U. Storch.
On regular sequences of binomials.
Manuscripta Math., 98(1):115-132, 1999. MR 1669583 (99k:13017) - 18.
- R. P. Stanley.
Relative invariants of finite groups generated by pseudoreflections.
J. Algebra, 49(1):134-148, 1977. MR 0460484 (57:477) - 19.
- B. Sturmfels.
Gröbner bases and convex polytopes, volume 8 of University Lecture Series.
American Mathematical Society, Providence, RI, 1996. MR 1363949 (97b:13034) - 20.
- A. Thoma.
Construction of set theoretic complete intersections via semigroup gluing.
Beiträge Algebra Geom., 41(1):195-198, 2000. MR 1745589 (2001h:14059) - 21.
- A. Thoma.
On the binomial arithmetical rank.
Arch. Math. (Basel), 74(1):22-25, 2000. MR 1728358 (2001a:14023) - 22.
- K. Watanabe.
Invariant subrings which are complete intersections. I. Invariant subrings of finite abelian groups.
Nagoya Math. J., 77:89-98, 1980. MR 0556310 (82d:13020) - 23.
- G. M. Ziegler.
Lectures on polytopes.
Springer, New York, 1995. MR 1311028 (96a:52011)
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Additional Information:
Eduardo
Cattani
Affiliation:
Department of Mathematics and Statistics, University of Massachusetts, Amherst, Massachusetts 01003
Email:
cattani@math.umass.edu
Raymond
Curran
Affiliation:
Department of Mathematics and Statistics, University of Massachusetts, Amherst, Massachusetts 01003
Address at time of publication:
Department of Mathematical and Computer Sciences, Metropolitan State College of Denver, Denver, Colorado 80202
Email:
rcurran@mscd.edu
Alicia
Dickenstein
Affiliation:
Departamento de Matematica, FCEyN, Universidad de Buenos Aires, (1428) Buenos Aires, Argentina
Email:
alidick@dm.uba.ar
DOI:
10.1090/S0002-9939-06-08513-3
PII:
S 0002-9939(06)08513-3
Received by editor(s):
January 11, 2005
Received by editor(s) in revised form:
August 18, 2005
Posted:
August 1, 2006
Additional Notes:
The first author was partially supported by NSF Grant DMS--0099707
The third author was partially supported by UBACYT X042, Argentina
Communicated by:
Michael Stillman
Copyright of article:
Copyright
2006,
American Mathematical Society
The copyright for this article reverts to public domain after 28 years from publication.
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