Complete Intersections in binomial and Lattice ideals

For the family of graded lattice ideals of dimension 1, we establish a complete intersection criterion in algebraic and geometric terms. In positive characteristic, it is shown that all ideals of this family are binomial set-theoretic complete intersections. In characteristic zero, we show that an arbitrary lattice ideal which is a binomial set-theoretic complete intersection is a complete intersection.

[1]  Juan José Salazar González,et al.  An algorithm for checking whether the toric ideal of an affine monomial curve is a complete intersection , 2007, J. Symb. Comput..

[2]  B. Sturmfels,et al.  Binomial Ideals , 1994, alg-geom/9401001.

[3]  Set-theoretic complete intersections on binomials , 2001 .

[4]  Rafael H. Villarreal,et al.  On systems of binomials in the ideal of a toric variety , 2002 .

[5]  John Little,et al.  CAYLEY-BACHARACH AND EVALUATION CODES ON COMPLETE INTERSECTIONS , 2003, math/0311129.

[6]  B. Sturmfels,et al.  Combinatorial Commutative Algebra , 2004 .

[7]  C. Delorme,et al.  Sous-monoïdes d’intersection complète de $N$ , 1976 .

[8]  Iwan M. Duursma,et al.  Reed-Muller Codes on Complete Intersections , 2001, Applicable Algebra in Engineering, Communication and Computing.

[9]  S. Eliahou.  Idéaux de définition des courbes monomiales , 1984 .

[10]  Rafael H. Villarreal,et al.  Monomial algebras and polyhedral geometry , 2001 .

[11]  Rafael H. Villarreal,et al.  Algebraic methods for parameterized codes and invariants of vanishing ideals over finite fields , 2010, Finite Fields Their Appl..

[12]  Rafael H. Villarreal,et al.  The minimum distance of parameterized codes on projective tori , 2010, Applicable Algebra in Engineering, Communication and Computing.

[13]  Jürgen Herzog,et al.  Generators and relations of abelian semigroups and semigroup rings , 1970 .

[14]  Johan P. Hansen,et al.  Linkage and Codes on Complete Intersections , 2003, Applicable Algebra in Engineering, Communication and Computing.

[15]  Apostolos Thoma,et al.  Binomial generation of the radical of a lattice ideal , 2008, 0811.3833.

[16]  Rafael H. Villarreal,et al.  Complete intersections in affine monomial curves , 2005 .

[17]  Walter D. Morris,et al.  Affine semigroup rings that are complete intersections , 1997 .

[18]  Edoardo Ballico,et al.  The Horace Method for Error-Correcting Codes , 2006, Applicable Algebra in Engineering, Communication and Computing.

[19]  R. Gilmer,et al.  Commutative Semigroup Rings , 1984 .

[20]  Apostolos Thoma.  On the set-theoretic complete intersection problem for monomial curves in An and Pn , 1995 .

[21]  W. Bruns,et al.  Cohen-Macaulay rings , 1993 .

[22]  Martin Kreuzer,et al.  Cayley-Bacharach schemes and their canonical modules , 1993 .

[23]  T. Moh.  Set-theoretic complete intersections , 1985 .

[24]  Apostolos Thoma,et al.  Complete intersection lattice ideals , 2004 .