For every d-dimensional Gorenstein polytope P of index r, the stringy E-polynomial E_st(P;u,v) vanishes if and only if P is thin, equivalently, its local h^-polynomial vanishes. If P is not thin, then E_st(P;u,v) has total degree 2(d+1-2r).

Batyrev–Nill stringy E-polynomial conjecture

The conjecture concerns when the stringy $E$ -polynomial of a Gorenstein polytope vanishes and what its degree should be. Johannes Knupfer and Benjamin Nill now claim the full criterion, the expected degree in the nonvanishing case, and a decomposition theorem for a broad class.

Known results

An earlier paper proved polynomiality of the stringy $E$ -function and vanishing when the associated Calabi–Yau dimension is negative, but did not establish the expected degree in the remaining cases.
A 2023 study characterized thin Gorenstein polytopes in every dimension and connected them with questions about stringy $E$ -polynomials.

September 2026 claimed resolution

Knupfer and Nill’s preprint claims that the stringy $E$ -polynomial vanishes exactly for thin Gorenstein polytopes and otherwise has degree $2(d+1-2r)$ , together with a decomposition theorem. The authors state that ChatGPT 5.6 Sol helped find the proofs; the preprint remains unrefereed, so the resolution is unverified.

Current status (as of September 2026): The conjecture is claimed solved by an unrefereed preprint, but the claimed equivalence, degree statement, and decomposition theorem remain independently unverified.

Batyrev–Nill stringy E-polynomial conjecture

Primary source

Additional references

Progress summary

Solutions 0