For a tuple of polynomials f=(f_1,…,f_r) and a monoid ideal K⊆N^r, let B_f^K⊆C[s_1,…,s_r] be the corresponding Bernstein–Sato ideal and let V(B_f^K)⊆C^r denote its zero locus. Budur's conjecture asserts that V(B_f^K) is a finite union of translates q+L, where q∈Q^r and L⊆C^r is a linear subspace defined over Q.

Budur’s translated-linear-subvariety conjecture for Bernstein–Sato ideals

Budur’s conjecture proposes that Bernstein–Sato ideals have a global zero-set structure built from translated linear pieces. Budur formulated it in 2012, with a published account in 2015, and established only partial results, including the hyperplane-arrangement case.

Known results

Budur (2012; published 2015) formulated the global and local conjectures and proved partial containment results.
Budur (2012; published 2015) verified the corresponding monodromy statement for hyperplane arrangements.
Later work listed by Budur includes zero-locus results from 2021 through 2025, but the retrieved descriptions do not state a disproof of this conjecture.

September 2026 counterexample claim

Wenzong Guo, Lei Wu, and Fanghan Xiang claim explicit formulas for several Bernstein–Sato ideals and exhibit a zero locus that is not a finite union of translated linear subvarieties, thereby disproving the proposed global structure theorem. This remains an unrefereed preprint result.

Current status (as of September 2026): The conjecture is claimed disproved by an unrefereed preprint, but the counterexample and its computations remain unverified.

Budur’s translated-linear-subvariety conjecture for Bernstein–Sato ideals

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Additional references

Progress summary

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