For every nonzero ideal I⊆C[x_1,…,x_n], every pole s_0 of a local topological zeta function associated with I is a root of the Bernstein–Sato polynomial b_I(s); equivalently, b_I(s_0)=0. The analogous assertion is also formulated for poles of the corresponding motivic zeta functions.

Strong monodromy conjecture

The strong monodromy conjecture asserts that poles of motivic or topological zeta functions should be roots of the relevant Bernstein–Sato polynomial. Public sources describe the general conjecture as wide open, despite substantial results for particular families.

Known results

Loeser proved the conjecture in dimension $2$ in $1988$ , apart from a non-reduced multiplicity issue; Blanco completed that case in $2024$ .
Reduced finite Coxeter hyperplane arrangements and several other arrangement families are known cases.
Maximal-minor determinantal ideals and sub-maximal Pfaffian ideals have explicit Bernstein–Sato formulas and satisfy the conjecture.

October 2026 claimed extension

On October 7, 2026, Quan Shi and Huaiqing Zuo’s unrefereed preprint A combinatorial criterion for detecting roots of Bernstein-Sato polynomials claimed the pole-to-root inclusion for determinantal and Pfaffian ideals. This is a restricted-family result, not a general solution, and its mathematical correctness has not been independently assessed.

Current status (as of October 2026): The conjecture is established in several special families, while the general statement remains open; the latest determinantal and Pfaffian claim is unverified.

Strong monodromy conjecture

Primary source

Additional references

Progress summary

Solutions 0