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Reciprocal domains and Cohen-Macaulay \(d\)-complexes in \(\mathbb R^ d\). (English) Zbl 1084.52515
Let \(C\) be a pointed polyhedral rational cone of full dimension \(d\).
R. P. Stanley has shown that, for two \(\Delta, \Delta'\) subcomplexes of \(C\) that are linearly separated reciprocal domains, their lattice point enumerators satisfy the relationship \(F_{C \setminus \Delta'} (x^{-1}) =(-1)^d F _{C \setminus \Delta} (x)\) [Adv. Math. 14, 194-253 (1974; Zbl 0294.05006)]. The authors show here that this relationship holds true if \(\Delta\) is a Cohen-Macaulay subcomplex over some field \(k\), a condition which is weaker than Stanley’s hypothesis. The paper also shows that if a \(d\)-dimensional proper subcomplex \(K\) of the boundary of a \((d+1)\)-polytope is Cohen-Macaulay over some field \(k\), then \(K\) has its topological space \(| K |\) isomorphic to a \(d\)-ball for \(d \leq 3\). This fails to be true for \(d=4\), as the authors show by using an example of Mazur coupled with a result by Shewchuk.

MSC:
52B20 Lattice polytopes in convex geometry (including relations with commutative algebra and algebraic geometry)
13H10 Special types (Cohen-Macaulay, Gorenstein, Buchsbaum, etc.)
05E99 Algebraic combinatorics
57Q05 General topology of complexes
51M20 Polyhedra and polytopes; regular figures, division of spaces
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