Reciprocal domains and {C}ohen-{M}acaulay {\(d\)}-complexes in {\(\mathbb R^ d\)} (Q1773145)

Let \(C\) be a pointed polyhedral rational cone of full dimension \(d\). \textit{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.