Supernormal Vector Configurations
Serkan Hoşten
, Diane MacLagan2
and Bernd Sturmfels2
2dagger
DOI: 10.1023/B:JACO.0000030705.93448.ce
Abstract
A configuration of lattice vectors is supernormal if it contains a Hilbert basis for every pointed cone spanned by a subset. We study such configurations from various perspectives, including triangulations, integer programming and Gröbner bases. Our main result is a bijection between virtual chambers of the configuration and virtual initial ideals of the associated binomial ideal.
Pages: 297–313
Keywords: triangulation; chamber complex; initial idea; Groebner Fan
Full Text: PDF
References
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2. L.J. Billera, I.M. Gel'fand, and B. Sturmfels, “Duality and minors of secondary polyhedra,” J. Combin. Theory Ser. B 57(2) (1993), 258-268.
3. J.A. de Loera, S. Ho\?sten, F. Santos, and B. Sturmfels, “The polytope of all triangulations of a point configuration,” Documenta Mathematica 1(4) (1996), 103-119 (electronic).
4. S. Ho\?sten and R.R. Thomas, “The associated primes of initial ideals of lattice ideals,” Mathematical Research Letters 6 (1999), 83-97.
5. S. Ho\?sten and D. Maclagan, “The vertex ideal of a lattice,” Advances in Applied Mathematics 29 (2002), 521-538.
6. M. Saito, B. Sturmfels, and N. Takayama, Gr\ddot obner Deformations of Hypergeometric Differential Equations, Springer, Heidelberg, 2000.
7. A. Schrijver, Theory of Linear and Integer Programming, John Wiley & Sons Ltd., Chichester, 1986.
8. B. Sturmfels, Gr\ddot obner Bases and Convex Polytopes, American Mathematical Society, Providence, RI, 1996.
9. B. Sturmfels and R.R. Thomas, “Variation of cost functions in integer programming,” Math. Programming 77(3, Ser. A) (1997), 357-387.
10. G. Ziegler, Lectures on Polytopes, Springer Verlag, Heidelberg, 1995.