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Buchstaber invariant of simple polytopes. (English. Russian original) Zbl 1184.52013

Russ. Math. Surv. 63, No. 5, 962-964 (2008); translation from Usp. Mat. Nauk 63, No. 5, 187-188 (2008).
With a simple \(n\)-polytope \(P\) with \(m\) facets can be associated a toric variety; the Buchstaber number \(s(P)\) is the largest dimension of a freely acting torus subgroup. It is known that \(1 \leq s(P) \leq m - n\), and that \(s(P) \geq m - \gamma(P)\), with \(\gamma(P)\) the chromatic number (for the facets) of \(P\). It is also known that \(s(P)\) is not determined by the \(f\)-vector of \(P\) and \(\gamma(P)\).
After listing various facts about \(s(P)\), the author here gives specific examples of this indeterminacy with \(m - n = 3\), working with Gale diagrams (though not named here as such). The calculations of how the \(f\)-vector (or, rather, \(h\)-vector) changes under bistellar operations are essentially those by the reviewer in [Isr. J. Math. 9, 559–570 (1971; Zbl 0209.53701)].

MSC:

52B05 Combinatorial properties of polytopes and polyhedra (number of faces, shortest paths, etc.)

Citations:

Zbl 0209.53701
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