Abstract
Recent work has focused on the roots $z\in\mathbb{C}$ of the Ehrhart polynomial of a lattice polytope $P$. The case when $\Re{z}=-1/2$ is of particular interest: these polytopes satisfy Golyshev's "canonical line hypothesis". We characterise such polytopes when $\mathrm{dim}(P)\leq 7$. We also consider the "half-strip condition", where all roots $z$ satisfy $-\mathrm{dim}(P)/2\leq\Re{z}\leq \mathrm{dim}(P)/2-1$, and show that this holds for any reflexive polytope with $\mathrm{dim}(P)\leq 5$. We give an example of a $10$-dimensional reflexive polytope which violates the half-strip condition, thus improving on an example by Ohsugi–Shibata in dimension $34$.
Highlights
Let P ⊂ Zd⊗ZQ be a convex lattice polytope of dimension d
This corresponds to a terminal Gorenstein fake weighted projective space
In higher dimensions, where the number of reflexive polytopes is vast, this becomes an essential tool for studying their classification
Summary
Let P ⊂ Zd⊗ZQ be a convex lattice polytope of dimension d. It was conjectured in [22] that for any reflexive polytope P , the roots z of LP satisfy the half-strip condition. Inequality (HS) holds when d 5, or for arbitrary d when P is real This result halves the bounds of (S) in the case of reflexive polytopes. This corresponds to a terminal Gorenstein fake weighted projective space. In higher dimensions, where the number of reflexive polytopes is vast, this becomes an essential tool for studying their classification
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have