YURAN XIU

math.CO — Combinatorics

An Arbor Polytope with Nonreal Ehrhart Roots

Contributed by YURAN XIU

We construct a singleton-labeled arbor on 243 vertices whose Ehrhart polynomial has a nonreal root. This disproves the conjecture that the Ehrhart roots of every arbor polytope are real and lie in $[-1,0]$. The polynomial factors as $(m+1)^{230}(7m+2)^8 f(m)/6144$, where $f$ is an explicit quintic. We derive this identity by counting lattice points for every nonnegative integer dilation, using three moments of subtree deficits. A negative value of $f'^2-ff''$ at $-3/10$ gives an exact nonreal-root certificate. The lattice-count identity and the resulting counterexample have been formalized and checked in Lean.