Multicomplexes and polynomials with real zeros

Abstract

We show that each polynomial a (z) = 1 + a1 z + ⋯ + ad zd in N [z] having only real zeros is the f-polynomial of a multicomplex. It follows that a (z) is also the h-polynomial of a Cohen-Macaulay ring and is the g-polynomial of a simplicial polytope. We conjecture that a (z) is also the f-polynomial of a simplicial complex and show that the multicomplex result implies this in the special case that the zeros of a (z) belong to the real interval [- 1, 0). We also show that for fixed d the conjecture can fail for at most finitely many polynomials having the required form. © 2006 Elsevier B.V. All rights reserved.