Geodesic
The root distributions of Ehrhart polynomials of free sums of reflexive polytopes
Masahiro Hachimori ; Akihiro Higashitani1 ; Yumi Yamada2
1Department of Pure and Applied Mathematics, Graduate School of Information Science and Technology, Osaka University
2Department of Policy and Planning Sciences, Graduate School of Systems and Information Engineering, Tsukuba
The electronic journal of combinatorics, Tome 29 (2022) no. 3
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In this paper, we study the root distributions of Ehrhart polynomials of free sums of certain reflexive polytopes. We investigate cases where the roots of the Ehrhart polynomials of the free sums of $A_d^\vee$'s or $A_d$'s lie on the canonical line $\mathrm{Re}(z)=-\frac{1}{2}$ on the complex plane $\mathbb{C}$, where $A_d$ denotes the root polytope of type A of dimension $d$ and $A_d^\vee$ denotes its polar dual. For example, it is proved that $A_m^\vee \oplus A_n^\vee$ with $\min\{m,n\} \leq 1$ or $m+n \leq 7$, $A_2^\vee \oplus (A_1^\vee)^{\oplus n}$ and $A_3^\vee \oplus (A_1^\vee)^{\oplus n}$ for any $n$ satisfy this property. We also perform computational experiments for other types of free sums of $A_n^\vee$'s or $A_n$'s.
DOI : 10.37236/10795
Classification : 52B20, 26C10, 11H06

Masahiro Hachimori    ; Akihiro Higashitani  1   ; Yumi Yamada  2

1 Department of Pure and Applied Mathematics, Graduate School of Information Science and Technology, Osaka University
2 Department of Policy and Planning Sciences, Graduate School of Systems and Information Engineering, Tsukuba 
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     author = {Masahiro Hachimori and Akihiro Higashitani and Yumi Yamada},
     title = {The root distributions of {Ehrhart} polynomials of free sums of reflexive polytopes},
     journal = {The electronic journal of combinatorics},
     year = {2022},
     volume = {29},
     number = {3},
     doi = {10.37236/10795},
     zbl = {7567723},
     url = {http://geodesic.mathdoc.fr/articles/10.37236/10795/}
}
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Masahiro Hachimori; Akihiro Higashitani; Yumi Yamada. The root distributions of Ehrhart polynomials of free sums of reflexive polytopes. The electronic journal of combinatorics, Tome 29 (2022) no. 3. doi: 10.37236/10795

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