Roots of the Ehrhart polynomial of hypersimplices

The Ehrhart polynomial of the $d$-th hypersimplex $\Delta(d,n)$ of order $n$ is studied. By computational experiments and a known result for $d=2$, we conjecture that the real part of every roots of the Ehrhart polynomial of $\Delta(d,n)$ is negative and larger than $- \frac{n}{d}$ if $n \geq 2d$. In this paper, we show that the conjecture is true when $d=3$ and that every root $a$ of the Ehrhart polynomial of $\Delta(d,n)$ satisfies $-\frac{n}{d} < {\rm Re} (a) < 1$ if $4 \leq d \ll n$.Comment: 18 pages, 8 figure

Strong Koszulness of the toric ring associated to a cut ideal

A cut ideal of a graph was introduced by Sturmfels and Sullivant. In this paper, we give a necessary and sufficient condition for toric rings associated to the cut ideal to be Strongly Koszul.Comment: 10 pages, 3 figure

Centrally symmetric configurations of order polytopes

It is shown that the toric ideal of the centrally symmetric configuration of the order polytope of a finite partially ordered set possesses a squarefree quadratic initial ideal. It then follows that the convex polytope arising from the centrally symmetric configuration of an order polytope is a normal Gorenstein Fano polytope.Comment: 9 pages, Proof of Theorem 2.2 is simplified. Major revision on Section

Perfectly contractile graphs and quadratic toric rings

Perfect graphs form one of the distinguished classes of finite simple graphs. In 2006, Chudnovsky, Robertson, Saymour and Thomas proved that a graph is perfect if and only if it has no odd holes and no odd antiholes as induced subgraphs, which was conjectured by Berge. We consider the class ${\mathcal A}$ of graphs that have no odd holes, no antiholes and no odd stretchers as induced subgraphs. In particular, every graph belonging to ${\mathcal A}$ is perfect. Everett and Reed conjectured that a graph belongs to ${\mathcal A}$ if and only if it is perfectly contractile. In the present paper, we discuss graphs belonging to ${\mathcal A}$ from a viewpoint of commutative algebra. In fact, we conjecture that a perfect graph $G$ belongs to ${\mathcal A}$ if and only if the toric ideal of the stable set polytope of $G$ is generated by quadratic binomials. Especially, we show that this conjecture is true for Meyniel graphs, perfectly orderable graphs, and clique separable graphs, which are perfectly contractile graphs.Comment: 10 page

マイクログリア依存的な成体脳シナプス再編成

学位の種別: 課程博士審査委員会委員 : （主査）東京大学教授 池谷 裕二, 東京大学教授 後藤 由季子, 東京大学准教授 八代田 英樹, 東京大学准教授 名黒 功, 東京大学准教授 小山 隆太University of Tokyo(東京大学

Smooth Fano polytopes whose Ehrhart polynomial has a root with large real part

The symmetric edge polytopes of odd cycles (del Pezzo polytopes) are known as smooth Fano polytopes. In this paper, we show that if the length of the cycle is 127, then the Ehrhart polynomial has a root whose real part is greater than the dimension. As a result, we have a smooth Fano polytope that is a counterexample to the two conjectures on the roots of Ehrhart polynomials.Comment: 4 pages, We changed the order of the auhors and omitted a lot of parts of the paper. (If you are interested in omitted parts, then please read v1