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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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