For every lattice polytope P, the associated q-Ehrhart series E_P(q,t) is a rational function and satisfies the corresponding q-reciprocity relation; moreover, the associated harmonic algebra is finitely generated, together with the further structural properties asserted in the Reiner–Rhoades formulation.

Reiner–Rhoades q-Ehrhart conjecture

Reiner and Rhoades formulated the conjecture in July 2024, predicting rationality and reciprocity for certain graded lattice-point series, together with finite-generation and related properties of a harmonic algebra.

Known results

Cavey (August 2025) gave lattice-triangle counterexamples to finite generation of the harmonic algebra, disproving that part for general lattice polytopes.
The same work left rationality of the $q$ -Ehrhart series open.
A March 2026 preprint claims the rationality, reciprocity, finite-generation, and Cohen–Macaulay assertions for unimodular zonotopes.

August 2026 hypersimplex-related result

A preprint reports generators for the orbit harmonics ideal and the required rationality and reciprocity statements for hypersimplex-related polytopes, claiming a resolution in that restricted class. The result is not yet peer reviewed, and no source establishes the conjecture for arbitrary lattice polytopes.

Current status (as of August 2026): The conjecture is claimed solved for hypersimplex-related polytopes, while finite generation is false for general lattice polytopes and the general rationality question remains open.

Reiner–Rhoades q-Ehrhart conjecture

Primary source

Additional references

Progress summary

Solutions 0