Let f=f_1… f_r with each f_k∈C[x_1,…,x_n], and set F=(f_1,…,f_r). Let mu:Y→ X be a log resolution of f. Write E_i_i∈ S for the irreducible components of f circmu, let a_i,j be the order of vanishing of f_j along E_i, let k_i be the order of vanishing of the determinant of the Jacobian of mu along E_i, and for I⊆ S define E_I^circ=bigcap_i∈ IE_isetminus bigcup_i∈ S setminus IE_i. The topological zeta function is Z_F^operatornametop(S)=∑_I⊆ Schi(E_I^circ)∏_i∈ Ifrac1a_i,1s_1+…+a_i,rs_r+k_i+1, and B_F is the Bernstein–Sato variety. Topological multivariable strong monodromy conjecture. The polar locus of Z_F^operatornametop(S) is contained in V(B_F). This conjecture relates poles of the resolution-theoretic topological zeta function to Bernstein–Sato varieties. The supplied text states the conjecture but gives no resolution evidence, so its status is recorded as open.

Topological multivariable strong monodromy conjecture

About 14 years old · traced to 2012

Let $f=f_{1}\cdots f_{r}$ with each $f_{k}\in\mathbb{C}[x_{1},\dots,x_{n}]$ , and set $F=(f_{1},\dots,f_{r})$ . Let $\mu:Y\to X$ be a log resolution of $f$ . Write $\{E_{i}\}_{i\in S}$ for the irreducible components of $f\circ\mu$ , let $a_{i,j}$ be the order of vanishing of $f_{j}$ along $E_{i}$ , let $k_{i}$ be the order of vanishing of the determinant of the Jacobian of $\mu$ along $E_{i}$ , and for $I\subseteq S$ define

E_{I}^{\circ}=\bigcap_{i\in I}E_{i}\setminus\bigcup_{i\in S\setminus I}E_{i}.

The topological zeta function is

Z_{F}^{\operatorname{top}}(S)=\sum_{I\subseteq S}\chi(E_{I}^{\circ})\prod_{i\in I}\frac{1}{a_{i,1}s_{1}+\cdots+a_{i,r}s_{r}+k_{i}+1},

and $B_{F}$ is the Bernstein–Sato variety.

Topological multivariable strong monodromy conjecture. The polar locus of $Z_{F}^{\operatorname{top}}(S)$ is contained in $V(B_{F})$ .

This conjecture relates poles of the resolution-theoretic topological zeta function to Bernstein–Sato varieties. The supplied text states the conjecture but gives no resolution evidence, so its status is recorded as open.

References

Primary source

Daniel Bath, “Bernstein-Sato Varieties and Annihilation of Powers”, arXiv:1907.05301 (2020).

Additional references

2 papers in this index state this conjecture (2012–2019). The statement above is taken from the most recent of them; the others are arXiv:1209.3725.

bernstein-sato-theory monodromy-conjectures topological-zeta-functions

Posted 1 month ago

Progress summary

Refreshed 1 month ago

Claimed progress

A new unrefereed preprint claims the conjecture for plane curves, while the general multivariable question remains open.

The conjecture asserts that every actual pole of the multivariable topological zeta function belongs to the zero set of the associated Bernstein–Sato ideal. For tuples with $p>1$ , the conjecture was posed in an earlier reference, but the supplied sources do not identify its author or date.

Known results

Tame hyperplane arrangements, including non-reduced arrangements: the conjecture holds by a 2019 result using work of Budur.
Walther proved the $r=1$ case for tame, indecomposable central arrangements.
The conjecture is known for tuples factorizing a tame hyperplane arrangement and for tuples of linear polynomials.
In the single-polynomial plane-curve case, Loeser proved the result apart from the non-reduced order/multiplicity case in 1988; Blanco completed that case in 2024.

August 2026 plane-curve claim

An August 26, 2026 report points to the preprint The Multivariable Strong Monodromy Conjecture for Plane Curves, which claims that every actual polar hyperplane lies in the zero locus of the Bernstein–Sato ideal for plane curves. This is progress on a restricted class, not a solution of the full conjecture, and remains unverified.

Current status (as of August 2026): the conjecture is established in several special cases and claimed for plane curves by an unrefereed preprint, but the general multivariable statement remains open.

Sources

No solutions have been posted yet.