On relations for zeros of $f$-polynomials and $f^{+}$-polynomials

TL;DR

This paper investigates the zeros of $f$-polynomials and $f^{+}$-polynomials associated with cluster complexes of root systems, revealing their real, simple roots within (0,1) and their interlacing properties.

Contribution

It establishes the real, simple roots of $f$- and $f^{+}$-polynomials for root systems and describes their interlacing, including asymptotic behavior for types A, B, and D.

Findings

$f$-polynomials have exactly $l$ real roots in (0,1)

$f^{+}$-polynomials have roots in (0,1] with a root at 1

Roots of $f^{+}$-polynomials interlace with roots of $f$-polynomials

Abstract

Let $\Phi$ be an irreducible (possibly noncrystallographic) root system of rank $l$ of type $P$. For the corresponding cluster complex $\Delta(P)$, which is known as pure $(l-1)$-dimensional simplicial complex, we define the generating function of the number of faces of $\Delta(P)$ with dimension $i-1$, which is called the {\it $f$-polynomial}. We show that the $f$-polynomial has exactly $l$ simple real zeros on the interval $(0, 1)$ and the smallest root for the infinite series of type $A_l$, $B_l$ and $D_l$ monotone decreasingly converges to zero as the rank $l$ tends to infinity. We also consider the generating function (called the {\it $f^{+}$-polynomial}) of the number of faces of the positive part $\Delta_{+}(P)$ of the complex $\Delta(P)$ with dimension $i-1$, whose zeros are real and simple and are located in the interval $(0, 1]$, including a simple root at $t=1$. We show that the roots $\{ t^{+}_{P, \nu+1} \}_{\nu=1}^{l-1}$ in decreasing order of $f^{+}$-polynomial alternate with the roots $\{ t_{P, \nu} \}_{\nu=1}^{l}$ in decreasing order of $f$-polynomial.