Toric ideals of Minkowski sums of unit simplices

Akihiro Higashitani,Hidefumi Ohsugi

doi:10.5802/alco.117

Abstract

In this paper, we discuss the toric ideals of Minkowski sums of unit simplices. More precisely, we prove that the toric ideal of Minkowski sum of unit simplices has a squarefree initial ideal and is generated by quadratic binomials. Moreover, we also prove that Minkowski sums of unit simplices have the integer decomposition property. Those results are a partial contribution to Oda conjecture and B{\o}gvad conjecture.

Highlights

Let P ⊂ Rd be a lattice polytope, which is a convex polytope all of whose vertices belong to the standard lattice Zd, of dimension d
Has the integer decomposition property if for any positive integer n and α ∈ nP ∩ Zd, there exist α1, . . . , αn ∈ P ∩ Zd such that α = α1 + · · · + αn
The toric ideal IA of A is the defining ideal of the toric ring K[A], i.e. it is the kernel of a surjective ring homomorphism π : K[x1, . . . , xm] → K[A] defined by π(xi) = tai

Summary

Introduction

The toric ideal of every smooth polytope is generated by quadratic binomials. There are many examples of lattice polytopes which are IDP but their toric ideals are not generated by quadratic binomials, and vice versa. It is proved in [6, Proposition 6.3] that for a given {yI }, we see that PnY ({yI }) = PnZ ({zI }) by setting zI = J⊂I yJ .

Results

Conclusion