Oda's conjecture for reflexive polytopes: some special cases
Binnan Tu
TL;DR
This paper advances Oda's conjecture by proving new IDP results in concrete settings. It shows that if $P$ is an $n$-dimensional simplicial reflexive polytope with each facet containing at most $n+1$ lattice points and admits a unimodular triangulation, while $Q$ is a lattice polytope with $\mathcal{V}(Q) \subset \mathcal{V}(P) \cup \{0\}$, then $(P,Q)$ is an IDP pair, i.e., $(P+Q) \cap \mathbb{Z}^n = (P\cap \mathbb{Z}^n) + (Q\cap \mathbb{Z}^n)$. It further proves that any two facet unimodular polytopes with at most two non-zero entries in each facet-normal row form IDP pairs, and that reflexive polytopes forming almost co-unimodular pairs also have the IDP property. These results connect unimodular triangulations, facet-normal matrices, and toric geometry, providing partial progress toward resolving Oda's conjecture under structural constraints.
Abstract
In this paper, we show that Oda's question holds for $n$-dimensional simplicial reflexive polytope $P$ and lattice polytope $Q$ containing the origin, when the vertex of $Q$ is either a vertex of $P$ or the origin, provided that $P$ has no more than $n+1$ lattice points on each facet and possesses unimodular triangulation. Then we prove Oda's question is true for any two facet unimodular polytopes whose matrix defining the facets has at most two non-zero entries in each row, and also true for any almost co-unimodular pair of reflexive polytopes.