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# Associate Professor Akihiro Higashitani

Area and Subject Taught | Algebraic Combinatorics |
---|---|

Research Theme(s) | Combinatorics on lattice polytopes and combinatorial commutative algebra |

Academic Degrees | Doctor (Science), Osaka University |

Keywords for Research Field | lattice polytopes, Ehrhart polynomial, graded commutative algebra, toric ring |

Office Phone Number | Not Public |

## Research Overview

A polygon is called a lattice polygon if all of its vertices are lattice points (integer points). A well-known formula “Pick’s Formula” can compute the area of a lattice polygon by counting the lattice points contained in it. Actually, there is an analogy of Pick’s formula for higher dimensional polygons, i.e., for lattice polytopes. Although such a formula looks complicated, we can explain it by the theory of “Ehrhart polynomials”. I’m interested in Ehrhart theory and I work on the study of the combinatorial structure of lattice polytopes by discussing their Ehrhart polynomials.

On the other hand, I also study graded commutative rings arising from combinatorial objects. In particular, I’m interested in “toric rings”, which contain the class of graded commutative algebras defined from lattice polytopes (called the Ehrhart ring).

In addition, I’m also interested in the algebraic geometry of “toric varieties” associated with lattice polytopes. We can study it by discussing the combinatorial structure of lattice polytopes.

## Notable Publications and Works in the Last Three Years

- Akihiro Higashitani, Minkowski sum of polytopes and its normality, Journal of Mathematical Society of Japan, 68 (2016), 1147-1159.
- Akihiro Higashitani, Almost Gorenstein homogeneous rings and their h-vectors, Journal of Algebra, 456 (2016), 190-206.
- Akihiro Higashitani, Smooth Fano polytopes arising from finite directed graphs, Kyoto Journal of Mathematics, 55 (2015), 579-592.
- Takayuki Hibi, Akihiro Higashitani, Lukas Katthän and Ryota Okazaki, Toric rings arising from cyclic polytopes, Communications in Algebra, 43 (2015), 778-794.