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

Let $\mathrm{\Phi }$ be an irreducible (possibly noncrystallographic) root system of rank $l$ of type $P$. For the corresponding cluster complex $\mathrm{\Delta }(P)$, which is known as pure $(l-1)$-dimensional simplicial complex, we define the generating function of the number of faces of $\mathrm{\Delta }(P)$ 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 ${\mathrm{\Delta }}_{+}(P)$ of the complex $\mathrm{\Delta }(P)$ 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 ${ t^{+}{P, \nu+1} }{\nu=1}^{l-1}$indecreasingorderof$f^{+}$-polynomial alternate with the roots $\{t_{P,\nu }{\}}_{\nu =1}^l$ in decreasing order of $f$-polynomial.