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Decomposing the secondary Cayley polytope. (English) Zbl 0957.52003
Regular triangulations and mixed subdivisions play an important role in algebraic geometry and are used in homotopy continuation methods for solving polynomial systems. The vertices of the secondary polytope (Gel’fand et al.) of a finite point configuration \(A\subset\mathbb{R}^d\) are in a one-to-one correspondence to the regular triangulations of \(A\).
The authors show that for (fine) mixed subdivisions of a tuple of point configurations, this secondary polytope can be Minkowski decomposed into nontrivial polytopes which are called mixed secondary polytopes and whose vertices correspond to regular-cell configurations (Sections 2-5). The connection between the edges of the mixed secondary polytopes and bistellar flips is explained in Section 6. Two examples are given in the final Section 7: for instance, if \(A_1=A_2= \cdots= A_n\) all are the same pair \(\{0,1\} \subset \mathbb{R}^1\), then the Cayley-embedding \(C(A_1, \dots, A_n)\) is the product of a line-segment and an \(n\)-simplex, and the secondary polytope is known to be the \(n\)-dimensional permutohedron, which is the Minkowski sum of \(n(n-1)/2\) line-segments.

52B12 Special polytopes (linear programming, centrally symmetric, etc.)
52B11 \(n\)-dimensional polytopes
51M20 Polyhedra and polytopes; regular figures, division of spaces
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