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Multiplicative Thom–Sebastiani property for Bernstein–Sato polynomials of ideals

Let a⊆C[x1,…,xm]\mathfrak a\subseteq \mathbb C[x_1,\ldots,x_m] and b⊆C[y1,…,yn]\mathfrak b\subseteq \mathbb C[y_1,\ldots,y_n] be nonzero ideals, and regard them as ideals in C[x1,…,xm,y1,…,yn]\mathbb C[x_1,\ldots,x_m,y_1,\ldots,y_n] by extension. Is the Bernstein–Sato polynomial of their product equal to the product of their Bernstein–Sato polynomials, namely bab(s)=ba(s)bb(s)b_{\mathfrak a\mathfrak b}(s)=b_{\mathfrak a}(s)b_{\mathfrak b}(s)?

References

Primary source

arXiv

Additional references

algebraic-geometrybernsteinsato-polynomialssingularity-theory
Posted

Progress summary

Refreshed
Claimed solved

A new unrefereed preprint claims to settle the ideal version of this structural question, but the claim has not been independently checked.

The problem asks whether a structural multiplication rule known for single functions extends to Bernstein–Sato data attached to ideals. The underlying function questions were attributed to Budur and Popa.

Known results

  • Lee (2024) and independently Shi–Zuo proved the function formula bfg(s)=bf(s)bg(s)b_{f g}(s)=b_f(s)b_g(s).
  • A 2024 preprint showed the unrestricted ideal analogue is false, even after ignoring root multiplicities, but proved it when one ideal is principal.
  • For monomial ideals, the same preprint obtained further inclusions and congruence results.
  • Przybyszewski Albarracín (May 2025) reported extensions for several Bernstein–Sato ideals, including product and concatenation formulas, without independently settling the full formulation.

2026 preprint

A new preprint claims that the multiplicative Thom–Sebastiani property for Bernstein–Sato polynomials of ideals is proved, thereby settling the tracked open questions. This is an unrefereed claim and has not been independently verified in the retrieved sources.

Current status (as of August 2026): The function case and several restricted ideal cases are established, while the full ideal statement is only claimed solved by an unrefereed 2026 preprint.

Sources

Solutions 0

No solutions have been posted yet.