Erokhovets, N. Yu 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)]. Reviewer: Peter McMullen (London) Cited in 1 ReviewCited in 11 Documents MSC: 52B05 Combinatorial properties of polytopes and polyhedra (number of faces, shortest paths, etc.) Keywords:simple polytope; torus action; Buchstaber invariant; chromatic number; \(f\)-vector; Gale diagram Citations:Zbl 0209.53701 PDFBibTeX XMLCite \textit{N. Y. Erokhovets}, Russ. Math. Surv. 63, No. 5, 962--964 (2008; Zbl 1184.52013); translation from Usp. Mat. Nauk 63, No. 5, 187--188 (2008) Full Text: DOI arXiv