Multicomplexes and polynomials with real zeros

We show that each polynomial a(z)=1+a1z+⋯+adzda(z)=1+a1z+⋯+adzd in N[z]N[z] having only real zeros is the f  -polynomial of a multicomplex. It follows that a(z)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)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)a(z) belong to the real interval [-1,0)[-1,0). We also show that for fixed d the conjecture can fail for at most finitely many polynomials having the required form.