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An Arbor Polytope with Nonreal Ehrhart Roots

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Contributed by YURAN XIU ↗

Version 1 / Oct 14, 2026 / CC BY 4.0

Abstract

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][-1,0]. The polynomial factors as (m+1)230(7m+2)8f(m)/6144(m+1)^{230}(7m+2)^8 f(m)/6144, where ff 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′′f'^2-ff'' at −3/10-3/10 gives an exact nonreal-root certificate. The lattice-count identity and the resulting counterexample have been formalized and checked in Lean.

References

  1. F. Chapoton, On posets and polytopes attached to arbors , arXiv:2503.04247v2 (https://arxiv.org/abs/2503.04247v2) (2025).arXiv
  2. C. A. Athanasiadis and F. Chapoton, Polytopes and posets associated to preorders , arXiv:2605.26916v1 (https://arxiv.org/abs/2605.26916v1) (2026).arXiv
  3. A. Higashitani, Counterexamples of the conjecture on roots of Ehrhart polynomials , arXiv:1106.4633 (https://arxiv.org/abs/1106.4633) (2011).arXiv

Version history

  1. v1Submitted by YURAN XIUInitial depositCurrentOct 14, 2026