Winfried Bruns, Gubeladze Joseph, Martin Henk, Alexander Martin, Robert Weismantel
A counterexample to an integer analogue of Carathéodory's theorem
The paper is published:
J. Reine Angew. Math. 510 (1999), 179-185
- MSC:
- 52B20 Lattice polytopes (including relations with commutative algebra and algebraic geometry), See also {06A08, 13F20, 13Hxx}
- 90C10 Integer programming
Abstract: For $n\geq 6$ we provide a counterexample to the conjecture that every
integral vector of a $n$-dimensional integral polyhedral pointed cone
$C$ can be written as a nonnegative integral combination of at most $n$
elements of the Hilbert basis of $C$. In fact, we show that in
general at least $\lfloor 7/6 \cdot n \rfloor$ elements of the
Hilbert basis are needed. (Also see the preprint "Normality and covering properties of affine semigroups" by W. Bruns and J. Gubeladze.)
Keywords: finitely generated rational cone, unimodular Hilbert cover, integral Carathéodory property, totally dual integtal linear system