For every tuple F=(f_1,…,f_r) of nonzero holomorphic germs on a smooth complex germ (X,0), every actual polar hyperplane H of the local multivariable topological zeta function Z^top_F,0(s_1,…,s_r) is contained in the zero locus of the Bernstein--Sato ideal B_F,0; equivalently, H⊆ V(B_F,0)⊆C^r.

Multivariable Strong Monodromy Conjecture

The conjecture predicts that actual polar hyperplanes of a tuple’s local multivariable topological zeta function lie in the zero locus of the corresponding Bernstein–Sato ideal. The new result concerns tuples of reduced plane curve germs, not arbitrary tuples of holomorphic germs.

Known results

Tame hyperplane-arrangement tuples satisfy the multivariable inclusion $\operatorname{Pol}(Z_F^{\mathrm{top}})\subset V(B_F)$ (Budur, 2019).
The conjecture is also known for tuples of linear polynomials and for tuples factorizing a tame hyperplane arrangement.

August 2026 plane-curve preprint

The preprint The Multivariable Strong Monodromy Conjecture for Plane Curves claims that every actual polar hyperplane for a tuple of reduced plane curve germs lies in the zero locus of its Bernstein–Sato ideal, thereby claiming the plane-curve case; this remains unverified.

Current status (as of August 2026): The plane-curve case is claimed settled by an unrefereed preprint, while the conjecture for arbitrary tuples of holomorphic germs remains open.

Multivariable Strong Monodromy Conjecture

Primary source

Additional references

Progress summary

Solutions 0