55 problems

Yano's conjecture. For generic curves in some $\Gamma$-constant deformation of $f$, the $b$-exponents are

Monodromy Conjecture. If $s_0$ is a pole of $Z_{f,W}^{\mathrm{mot}}(s)$, then $(s-s_0)\mid b_f(s)$, and $e^{s_0\cdot 2\pi \sqrt{-1}}$ is an eigenvalue of a local monodromy at some…

Bitoun–Schedler's conjecture. The length of ${\mathcal D}_Xf^{-1}$ should be

Let $d=(d_K)_{K\in{[m]\choose n}}$ be the tuple of maximal minors of a generic $m\times n$ matrix. For each $\alpha\in\mathbb{Z}$, let $F_{\alpha}$ be the $\mathcal{D}$-module asso…

Let $f$ be the defining polynomial of the hyperplane arrangement, let $\lambda=\exp(-2\pi i\alpha)$, and let $J(\lambda)$ denote the set introduced in the paper. Suppose condition…

The bound conjecture. It may be conjectured that

Let $X$ be the smooth variety and let $f_1,\dots,f_r\in \mathscr{O}_X(X)$ be a regular sequence. Set $g=f_1\cdots f_r$ and let $U$ denote the open subset on which the relevant rest…

Extension conjecture. The sharp bound under the first hypothesis, including Proposition 1 and Theorem 2, should extend to the case where $X$ has only $A_2$-singularities instead of…

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…

Let $s_0$ be a pole of the motivic zeta function, and let $k$ be its pole order. Stronger Monodromy Conjecture. The number $s_0$ is a root of the Bernstein--Sato polynomial with mu…

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 a…

Let $D$ be the ring of differential operators, let $f$ and $g$ be the functions under consideration, and let $I(s_1,s_2)$ be the ideal defined in the paper. Write $I(s,s)$ for its…

Let $f\in k[x_1,\ldots,x_n]$, where $k=\mathbb R$ or $\mathbb C$, and let $Z_{f,\varphi}(s)$ be the associated complex version of the Archimedean zeta function. Let $b_f(s)$ denote…

Let $D$ be a linear free divisor, and let $b_D(s)$ denote its $b$-function. Granger–Schulze conjecture. The roots of $b_D(s)$ are symmetric about $-1$. This is proved in the paper…

Monodromy conjecture. The poles of $Z_f^{mot}(s)$ are roots of $b_f(s)$.

Let $X$ be the ambient space, let $f_1,\dots,f_r\in \mathscr{O}_X(X)$ be a regular sequence, and let $g$ and $U$ be as in the preceding construction. Write $b_g(s)$ for the Bernste…

The indecomposability criterion for the $n/d$-conjecture. If, for every nontrivial decomposition $(\mathbb{C}^{n})^{*}=W_{1}\oplus W_{2}$ and every integer $0\leq k\leq d$ satisfyi…

Budur–Mustaţă–Teitler's $n/d$-conjecture. The number $-\frac{n}{d}$ is a root of $b_{f,0}(s)$.

The differential-value conjecture. For any element $\lambda\in\Lambda_C\setminus\Gamma_C$, the rational number $-\lambda/nm$ is a root of the Bernstein–Sato polynomial of $C$.

Let $f\in\mathbb C[z_1,\ldots,z_n]\setminus\mathbb C$ be homogeneous of degree $d$, and suppose that the hypersurface $\{f=0\}$ is a central essential indecomposable hyperplane arr…

Let $f: X \longrightarrow \mathbb{C}$ be a non-constant regular function and let $x \in X$. Denote by $Z_{\textnormal{mot}, x}(f; s)$ the local motivic zeta function and by…

Multiplicative Thom–Sebastiani conjecture. One has

Monodromy conjecture. If $s_0\in\mathbb C$ is a pole of $Z_{f,\operatorname{top}}(s)$, then $e^{2\pi i s_0}$ is an eigenvalue of $T_f$ and $s_0$ is a zero of $b_f(s)$.

Let $X$ be a smooth affine variety, let $Z$ be a closed subscheme defined by an ideal $I\subset\mathcal O_X$, and let $x\in Z$. Write $b_I(s)$ and $b_{I,x}(s)$ for the global and l…

Igusa zeta function pole conjecture. The poles of the Igusa zeta function, after subtracting a natural number, are contained in the set of roots of $b_f(s)$.