Interesting and challenging examples for Normaliz

 

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Each example is provided as a separate zip file containing input and output. The mathematical background and more information is provided by the references specified.

 

In connection with the Singular library, examples 3 and 6 (for 4x4x3) are also discussed in http://www.mathematik.uni-kl.de/ftp/pub/Math/Singular/utils/NmzLibApp.txt. The cooperation of Normaliz and a computer algebra system such as Singular or Macaulay 2 is especially useful for example 3.

 

Note: some of the examples require Normaliz 2.5.

 

  1. An example of a normal lattice polytope that has no unimodular cover and even violates the integral Carathéodory property [BG, Section 2.D], [BG2], [BGHMW].

 

  1. An example of a lattice polytope satisfying ICP, but without unimodular cover [B].

 

  1. An example of a non-normal very ample lattice polytope [BG1]. (See [BG, p. 87] for further examples.)

 

  1. A 3-dimensional tight cone [BG, Section 2.D].

 

  1. A nonnormal rectangular simplex of dimension 3 [BG0].

 

  1. The missing cases in the Ohsugi-Hibi classification of normal monoids derived from contingency tables (4x4x3, 5x4x3, 5x5x3) and the first nonnormal case (6x4x3); see [OH], [BHIKS]. With the exception of 4x4x3, these examples require Normaliz 2.5. The output for 4x4x3 was created in mode “Hilbert basis polynomial” (command line option –h). Make sure to use “Hilbert basis”  (command line option –N) for the other cases. The largest example 5x5x3 takes about 15 hours on a single processor machine.

 

  1. The semi-graphoid for N = 5. (requires “Hilbert basis”  (command line option –N)) [HMSSW], [BHIKS].

 

  1. Verification of a conjecture of Sturmfels and Sullivant for some specific graphs [SS], [BHIKS].

 

  1. Two models of statistical ranking discussed in [SW], the inversion model for n = 6 (lo6) and the ascending model for n = 5 (bo5).

 

References

 

[B] W. Bruns,  On the integral Carathéodory property. Experiment. Math. 16 (2007), 359 - 365.

 

[BG] W. Bruns and J. Gubeladze. Polytopes, rings, and K-theory. Springer Monographs in Mathematics (2009).

 

[BG0] W. Bruns and J. Gubeladze. Rectangular simplicial semigroups. In Commutative algebra, algebraic geometry, and computational   methods, D. Eisenbud, Ed., Springer Singapore, 1999, pp. 201-214.

 

[BG1] W. Bruns and J. Gubeladze, Polytopal linear groups. J. Algebra 218 (1999), 715-737.

 

[BG2] W. Bruns and J. Gubeladze. Normality and covering properties of affine semigroups. J. Reine Angew. Math. 510 (1999), 151 - 178.

 

[BGHMW] W. Bruns, J. Gubeladze, M. Henk, A. Martin, and R. Weismantel. A counterexample to an integer analogue of Carathéodory's theorem. J. Reine Angew. Math. 510 (1999), 179 - 185.

 

[BHIKS] W. Bruns, R. Hemmecke, B. Ichim, M. Köppe, and C. Söger. Challenging computations of Hilbert bases of cones associated with algebraic statistics. Exp. Math., to appear. (Preprint arXiv:1001.4145.)

 

[HMSSW] R. Hemmecke, J. Morton, A Shiu, B Sturmfels, and O. Wienand. Three counterexamples on semigraphoids. Comb. Probab.  Comput.  17 (2008), 239 - 257.

 

[OH]  H.Ohsugi and T. Hibi. Toric ideals arising from contingency tables. In: Commutative Algebra and Combinatorics.  Ramanujan Mathematical Society Lecture Note Series 4 (2006), 87 - 111.

 

[SS] B. Sturmfels and S. Sullivant. Toric   geometry of cuts and splits. Mich. Math. J. 57 (2008), 689 - 709.

 

[SW] B. Sturmfels and V. Welker. Commutative Algebra of Statistical Ranking. arXiv:1101.1597