Zeros of rank-generating functions of Cohen-Macaulay complexes (Q1893998)

Many combinatorial polynomials are related to rank-generating functions of Cohen-Macaulay complexes; notable among these reliability, chromatic, flow, Birkhoff, and order polynomials. For a natural number \(j\) let \(x_{\langle j\rangle}= x(x- 1)\dots(x- j+ 1)\) denote the \(j\)th falling factorial polynomial, and define a linear transformation \(S: {\mathbf R}[x]\to {\mathbf R}[x]\) by \(Sx_{\langle j\rangle}= x^j\) and linear extension. The author proves the following main results: (1) Let \(p\in {\mathbf R}[x]\) be any polynomial, say \(p(x)= \sum^d_{i= 0} c_i x^i(x+ 1)^{d- i}\). If \(c_i\geq 0\) for all \(i= 0,\dots, d\) then \(Sp(x)\) has only real nonpositive zeros. (2) Let \(p\in {\mathbf R}[x]\) be a polynomial such that \(p(0)= 0\), say \(p(x)= x \sum^d_{i= 0} c_i x^i(x- 1)^{d- i}\). If \(c_i\geq 0\) for all \(i= 0,\dots, d\) then \(Sp(x)\) has only real nonpositive zeros. The author also discusses direct applications to the rank-generating functions of Cohen-Macaulay complexes and some consequences for each of the aforementioned classes of polynomials.