Oda's conjecture for reflexive polytopes: some special cases

TL;DR

This paper investigates Oda's conjecture for specific classes of reflexive polytopes, establishing its validity for certain simplicial and facet-unimodular cases, thus advancing understanding of the conjecture's scope.

Contribution

The paper proves Oda's conjecture for simplicial reflexive polytopes with limited lattice points per facet and for polytopes with facet matrices having at most two non-zero entries per row.

Findings

Oda's question holds for certain simplicial reflexive polytopes with unimodular triangulation.

Oda's question is true for facet-unimodular polytopes with specific matrix constraints.

The conjecture is valid for almost co-unimodular pairs of reflexive polytopes.

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.