Revista Matemática Iberoamericana

Minimal Resolutions of Lattice Ideals and Integer Linear Programming

Emilio Briales, Antonio Campillo, Pilar Pisón, and Alberto Vigneron

Full-text: Open access


A combinatorial description of the minimal free resolution of a lattice ideal allows us to the connection of Integer Linear Programming and Algebra. The non null reduced homology spaces of some simplicial complexes are the key. The extremal rays of the associated cone reduce the number of variables.

Article information

Rev. Mat. Iberoamericana, Volume 19, Number 2 (2003), 287-306.

First available in Project Euclid: 8 September 2003

Permanent link to this document

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 13D02: Syzygies, resolutions, complexes 14M25: Toric varieties, Newton polyhedra [See also 52B20]
Secondary: 13P10: Gröbner bases; other bases for ideals and modules (e.g., Janet and border bases) 68W30: Symbolic computation and algebraic computation [See also 11Yxx, 12Y05, 13Pxx, 14Qxx, 16Z05, 17-08, 33F10] 90C27: Combinatorial optimization

resolutions simplicial complex syzygy lattice ideal regularity integer linear programming Hilbert bases Gröbner bases


Briales, Emilio; Campillo, Antonio; Pisón, Pilar; Vigneron, Alberto. Minimal Resolutions of Lattice Ideals and Integer Linear Programming. Rev. Mat. Iberoamericana 19 (2003), no. 2, 287--306.

Export citation


  • Aramova, A. and Herzog, J.: Free resolution and Koszul homology. J. of Pure Appl. Algebra 105 (1995), no. 1, 1-16.
  • Bayer, D., Peeva, I. and Sturmfels, B.: Monomial resolutions. Math. Res. Lett. 5, (1998), no. 1-2, 31-46.
  • Bayer, D., Popescu, S. and Sturmfels, B.: Syzygies of unimodular Lawrence ideals. J. Reine Angew. Math. 534 (2001), 169-186.
  • Bayer, D. and Stillman, M.: A criterion for detecting m-regularity. Invent. Math. 87 (1987), no. 1, 1-11.
  • Bayer, D. and Sturmfels, B.: Cellular resolutions of monomial modules. J. Reine Angew. Math. 502 (1998), 123-140.
  • Bresinsky, H.: Binomial generating sets for monomial curves, with applications in $\mathbbA^4$. Rend. Sem. Mat. Univ. Politec. Torino 46 (1998), no. 3, 353-370 (1990).
  • Briales, E., Campillo, A., Marijuán, C. and Pisón, P.: Combinatorics of syzygies for semigroup algebras. Collect. Math. 49, (1998), no. 2-3, 239-256.
  • Briales, E., Campillo, A., Marijuán, C. and Pisón, P.: Minimal systems of generators for ideals of semigroups. J. Pure Appl. Algebra 124 (1998), no. 1-3, 7-30.
  • Briales, E., Campillo, A. and Pisón, P.: On the equations defining toric projective varieties. In Geometric and Combinatorial Aspects of Commutative Algebra (Messina, 1999), 57-66. Lecture Notes in Pure and Appl. Math. 217, Dekker, New York, 2001.
  • Briales, E., Campillo, A., Pisón, P. and Vigneron, A.: Simplicial complexes and syzygies of lattice ideals. In Symbolic Computation: Solving Equations in Algebra, Geometry and Engineering, 169-183, Contemp. Math. 286, Amer. Math. Soc., Providence, RI, 2001.
  • Briales, E., Pisón, P. and Vigneron, A.: Cotas para la Regularidad de un Ideal de Retículo. In Actas del Encuentro de Matemáticos Andaluces, 171-178. Servicio de publicaciones de la Univ. de Sevilla, 2002.
  • Briales, E., Pisón, P. and Vigneron, A.: The Regularity of a Toric Variety. J. Algebra 237 (2001), no. 1, 165-185.
  • Campillo, A. and Giménez, P.: Syzygies of affine toric varieties. J. Algebra 225 (2000), no. 1, 142-161.
  • Campillo, A. and Marijuán, C.: Higher order relations for a numerical semigroup. Sém. Théor. Nombres Bordeaux (2) 3 (1991), no. 2, 249-260.
  • Campillo, A. and Pisón, P.: Generators of a monomial curve and graphs for the associated semigroup. Bull. Soc. Math. Belg. Sér. A 45 (1993), no. 1-2, 45-58.
  • Campillo, A. and Pisón, P.: L'idéal d'un semi-groupe de type fini. C. R. Acad. Sci. Paris Sér. I Math. 316 (1993), no. 12, 1303-1306.
  • Campillo, A. and Pisón, P.: Toric Mathematics from semigroup viewpoint. In Ring Theory and Algebraic Geometry, 95-112. Lectures Notes in Pure and Applied Mathematics 221, Dekker, New York, 2001.
  • Clausen, M. and Fortenbacher, A.: Efficient solution of linear Diophantine equations. J. Symbolic Comput. 8 (1989), no. 1-2, 201-216.
  • Conti, P. and Traverso, C.: Buchberger algorithm and integer programming. In Applied algebra, algebraic algorithms and error-correcting codes, New Orleans, LA, 1991, 130-139. Lecture Notes in Comput. Sci., 539, Springer, Berlin, 1991.
  • Contejean, E. and Devie, H.: An efficient incremental algorithm for solving systems of linear Diophantine equations. Inform. and Comput. 113 (1994), no. 1, 143-172.
  • Cox, D.: Recent developments in toric geometry. In Algebraic geometry (Santa Cruz, 1995), 389-436. Proc. Sympos. Pure Math. 62, Part 2. Amer. Math. Soc., Providence, RI, 1997.
  • Di Biase, F. and Urbanke, R.: An algorithm to calculate the kernel of certain polynomial ring homomorphisms. Experiment. Math. 4 (1995), no. 3, 227-234.
  • Domenjoud, E.: Outils pour la Déduction Automatique dans les Théories Associatives-Commutatives. Thése de doctorat d'Université, Université de Nancy I, 1991.
  • Eisenbud, D. and Goto, S.: Linear free resolutions and minimal multiplicity. J. Algebra 88 (1984), no. 1, 89-133.
  • Eisenbud, D. and Sturmfels, B.: Binomial ideals. Duke Math. J. 84 (1996), no. 1, 1-45.
  • Fulton, W.: Introduction to Toric Varieties. Annals of Mathematics Studies 131. Princeton University Press, Princeton, 1993.
  • Gelfand, I. Kapranov, M. and Zelevinsky, A.: Discriminants, resultants, and multidimensional determinants. Mathematics: Theory & Applications, Birkhäuser, Boston, Inc., Boston, MA, 1994.
  • Graver, J. E.: On the foundations of linear and integer linear programming I. Math. Programming 9 (1975), no. 2, 207-226.
  • Herzog, J.: Generators and relations of abelian semigroups and semigroup rings. Manuscripta Math. 3 (1970), 175-193.
  • Hochster, M.: Cohen-Macaulay rings, combinatorics, and simplicial complexes. In Ring theory, II (Proc. Second Conf., Univ. Oklahoma, Norman, Okla., 1975), 171-223. Lect. Notes in Pure and Appl. Math. 26, Dekker, New York, 1977.
  • Hosten, S. and Sturmfels, B.: GRIN: an implementation of Gröbner bases for integer programming. In Integer programming and combinatorial optimization (Copenhagen, 1995), 267-276. Lecture Notes in Comput. Sci. 920, Springer, Berlin, 1995.
  • Kunz, E.: The value-semigroup of a one-dimensional Gorenstein ring. Proc. Amer. Math. Soc. 25 (1970), 748-751.
  • La Scala, R. and Stillman, M.: Strategies for computing minimal free resolutions. J. Symbolic Comput. 26 (1998), no. 4, 409-431.
  • Ojeda, I. and Piedra, R.: Cellular binomial ideals. Primary decomposition of binomial ideals. J. Symbolic Comput. 30 (2000), no. 4, 383-400.
  • Ojeda, I. and Piedra, R.: Index of nilpotency of binomial ideals. J. Algebra 255 (2002), no. 1, 135-147.
  • Ojeda, I. and Pisón, P.: The hull resolution of a monomial curve in $\mathbbA^3(k)$. Prepublicaciones del Departamento de Álgebra de la Universidad de Sevilla 12, 2001.
  • Papadimitriou, C. H.: On the complexity of integer programming. J. Assoc. Comput. Mach. 28 (1981), no. 4, 765-768.
  • Peeva, I. and Sturmfels, B.: Generic lattice ideals. J. Amer. Math. Soc. 11 (1998), no. 2, 363-373.
  • Peeva, I. and Sturmfels, B.: Syzygies of codimension 2 lattice ideals. Math. Z. 229 (1998), no. 1, 163-194.
  • Pisón, P.: Métodos combinatorios en Álgebra local y Curvas monomiales en dimensión 4. Doctoral thesis, Universidad de Sevilla, 1991.
  • Pisón, P.: The short resolution of a lattice ideal. Proc. Amer. Math. Soc., to appear.
  • Pisón-Casares, P. and Vigneron-Tenorio, A.: Computing Toric First Syzygies. International Conference IMACS-ACA, El Escorial, Madrid, Spain, 1999.
  • Pisón-Casares, P. and Vigneron-Tenorio, A.: First syzygies of toric varieties and Diophantine equations in congruence. Comm. Algebra 29 (2001), no. 4, 1445-1466.
  • Pisón-Casares, P. and Vigneron-Tenorio, A.: $\BbbN$-solutions to linear systems over $\BbbZ$. Prepublicación de la Universidad de Sevilla, Sección Álgebra 43, 1998.
  • Pisón-Casares, P. and Vigneron-Tenorio, A.: On the Graver Bases of Semigroup Ideals. Prepublicaciones del Departamento de Álgebra de la Universidad de Sevilla 10 (2001), 1-12, preprint.
  • Pottier, L.: Minimal solutions of linear diophantine systems: bounds and algorithms. In Rewriting techniques and applications (Como, 1991), 162-173. Lectures Notes in Comput. Sci. 488, Springer, Berlin, 1991.
  • Rosales, J. C.: Semigrupos numéricos. Doctoral thesis, Universidad de Granada, 1991.
  • Stanley, R.: Combinatorics and commutative algebra, second edition. Progress in Mathematics 41. Birkhäuser, Boston-Basel-Berlin, 1996.
  • Sturmfels, B.: Equations defining toric varieties. In Algebraic geometry (Santa Cruz, 1995), 437-449. Proc. Sympos. Pure Math. 62 Part 2, Amer. Math. Soc., Providence, RI, 1997.
  • Sturmfels, B.: Gröbner Bases and Convex Polytopes. University Lectures Series 8, American Mathematical Society, Providence, RI, 1995.
  • Sturmfels, B.: Gröbner bases of Toric Varieties. Tohoku Math. J. (2) 43 (1991), no. 2, 249-261.
  • Vigneron-Tenorio, A.: Semigroup ideals and linear Diophantine equations. Linear Algebra Appl. 295 (1999), no. 1-3, 133-144.
  • Vigneron-Tenorio, A.: Álgebras de Semigrupos y Aplicaciones. Doctoral thesis, Universidad de Sevilla, 2000.