Rationality of the q-Ehrhart series

Determine whether the q-Ehrhart series E_P(t,q) is a rational function of t and q for every lattice polytope P.

Background

The paper studies the harmonic algebra H_P, whose bigraded Hilbert series is the q-Ehrhart series E_P(t,q). Reiner and Rhoades conjectured that H_P is finitely generated for every lattice polytope, which would imply rationality of E_P(t,q) in t and q.

The paper disproves the finite-generation portion of that conjecture by constructing lattice triangles with non-finitely generated harmonic algebras. However, the authors explicitly state that the weaker rationality assertion remains unresolved, so rationality is included as a separate open problem.

References

The weaker conjecture, the rationality of the $q$-Ehrhart series $E_P(t,q)$, remains open.

— Graded Ehrhart theory and toric geometry  (2508.19176 - Cavey, 26 Aug 2025) in Introduction, paragraph following Conjecture 1 (the displayed conjecture labeled Conjecture \ref{conj})