The global and local Strong Monodromy Conjecture

Let $f \in \operatorname{\mathbb{C}}[x_1,\ldots,x_n] \setminus \mathbb{C}$ with $f(0)=0$. Write $Z_f^{\operatorname{mot}}(s)$ and $Z_{f,0}^{\operatorname{mot}}(s)$ for the global and local motivic zeta functions, and $b_f(s)$ and $b_{f,0}(s)$ for the corresponding global and local Bernstein--Sato polynomials. Strong Monodromy Conjecture. The poles of the motivic zeta functions are contained in the zero sets of the corresponding Bernstein--Sato polynomials:

$$\operatorname{Poles}(Z_f^{\operatorname{mot}}(s)) \subset \operatorname{Zeroes}(b_f(s)),$$

and

$$\operatorname{Poles}(Z_{f,0}^{\operatorname{mot}}(s)) \subset \operatorname{Zeroes}(b_{f,0}(s)).$$

The conjecture proposes a direct relationship between poles of motivic zeta functions and roots of Bernstein--Sato polynomials, both of which are singularity invariants. Its local form is related to the eigenvalues of the algebraic monodromy of Milnor fibers; the general conjecture remains open.