Igusa–Denef–Loeser Strong Monodromy Conjecture

Let $f$ be a polynomial and let $b_f(s)$ denote its Bernstein–Sato polynomial. For $f\in\mathbb{Q}[x_1,\ldots,x_n]$, let $Z_{f,p}(s)$ be its $p$-adic zeta function. For $f\in\mathbb{C}[x_1,\ldots,x_n]$, let $Z^{{\mathrm{global}}}_{f,\,\mathrm{top}}(s)$ be its global topological zeta function.

Strong Monodromy Conjecture.

1. If $f\in\mathbb{Q}[x_1,\ldots,x_n]$, then, for all but finitely many primes $p$, every pole $s_0$ of $Z_{f,p}(s)$ has $\operatorname{Re}(s_0)$ as a root of $b_f(s)$.
2. If $f\in\mathbb{C}[x_1,\ldots,x_n]$, then every pole $s_0$ of $Z^{{\mathrm{global}}}_{f,\,\mathrm{top}}(s)$ is a root of $b_f(s)$.

The conjecture predicts a relationship between poles of arithmetic and topological zeta functions and the Bernstein–Sato polynomial, linking arithmetic data with the geometry and monodromy of hypersurface singularities. The source proves the conjecture for hyperplane arrangements, while the general case remains open.