The Root Distributions of Ehrhart Polynomials of Free Sums of Reflexive Polytopes

In this paper, we study the root distributions of Ehrhart polynomials free sums certain reflexive polytopes. We investigate cases where roots $A_d^\vee$'s or $A_d$'s lie on canonical line $\mathrm{Re}(z)=-\frac{1}{2}$ complex plane $\mathbb{C}$, $A_d$ denotes polytope type A dimension $d$ and $A_d^\vee$ its polar dual. For example, it is proved that $A_m^\vee \oplus A_n^\vee$ with $\min\{m,n\} \leq 1$ $m+n 7$, $A_2^\vee (A_1^\vee)^{\oplus n}$ $A_3^\vee for any $n$ satisfy property. also perform computational experiments other types $A_n^\vee$'s $A_n$'s.