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Challenging computations of Hilbert bases of cones associated with algebraic statistics. (English) Zbl 1273.13052

Summary: We present two independent computational proofs that the monoid derived from \(5\times 5\times 3\) contingency tables is normal, completing the classification by H. Hibi and T. Ohsugi [in: Commutative algebra and combinatorics. Part I: Computational algebra and combinatorics of toric ideals. Part II: Topics in commutative algebra and combinatorics. Proceedings of the international workshop and conference on computational algebraic geometry, Bangalore, India, December 8–13, 2003. Mysore: Ramanujan Mathematical Society. Ramanujan Mathematical Society Lecture Notes Series 4, 91–115 (2007; Zbl 1187.13024)]. We show that Vlach’s vector disproving normality for the monoid derived from \(6\times 4 \times 3\) contingency tables is the unique minimal such vector up to symmetry. Finally, we compute the full Hilbert basis of the cone associated with the nonnormal monoid of the semigraphoid for \(| N|=5\). The computations are based on extensions of the packages \(\mathtt LattE-4ti2\) and \(\mathtt Normaliz\).

MSC:

13P25 Applications of commutative algebra (e.g., to statistics, control theory, optimization, etc.)
14M25 Toric varieties, Newton polyhedra, Okounkov bodies
52B20 Lattice polytopes in convex geometry (including relations with commutative algebra and algebraic geometry)
14C05 Parametrization (Chow and Hilbert schemes)

Citations:

Zbl 1187.13024

Software:

Normaliz; 4ti2
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