Abstract
We present an elementary proof of a result due to Ewald and Wessels: in a pointed, polyhedral cone of dimension n ≥ 3 with integer-valued generators, any linearly independent generator representation for a minimal Hilbert basis element has coefficient sum less than n — 1. Our proof makes explicit use of the geometry of the polyhedron given by the convex hull of the Hilbert basis elements.
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References
GÜnter Ewald: personal communication, 1990.
GÜnter Ewald & Uwe Wessels: On the ampleness of invertible sheaves in complete projective toric varieties, Results in Mathematics 19 (1991), 275–278.
Jiyong LIU: Hilbert Bases with the Carathéodory Property, Ph. D. Thesis, Cornell University, Ithaca NY, 1991.
Jlyong LIU & Leslie E. Trotter, Jr.: Hilbert covers and partitions of rational, polyhedral cones, preprint, Cornell University 1991.
Alexander Schrijvr: Theory of Linear and Integer Programming, John Wiley and Sons, Chichester, 1986.
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This research was partially supported by NSF Grant DMS89-20550.
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Liu, J., Trotter, L.E. & Ziegler, G.M. On the Height of the Minimal Hilbert Basis. Results. Math. 23, 374–376 (1993). https://doi.org/10.1007/BF03322309
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DOI: https://doi.org/10.1007/BF03322309