25 problems

Spanning conjecture. If $k eleq r eleq 2k-n-2$, then $(Rn/(E+(Q-1)))r$ is spanned by the expressions in the generating family for which $i1<(n-1)+(r-k)$; if $k-n<r<k$, the expressi…

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…

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 $f \in \operatorname{\mathbb{C}}[x_1,\ldots,x_n] \setminus \mathbb{C}$ with $f(0)=0$. Write $Z_f^{\operatorname{mot}}(s)$ and $Z_{f,0}^{\operatorname{mot}}(s)$ for the global a…

Let $X$ 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 $C$.

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…

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…

Let $Z$ be a reduced local complete intersection of codimension $r$. Let $p(Z)$ denote its singularity level, and let $\widetilde{\alpha}(Z)$ be the negative of the largest root of…

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

Facet-defining conjecture. For every rigid ray $\tau\in \operatorname{Trop}(X)\cap\mathbb{R}_{\geq 0}^p$,

Let $X$ be a complex manifold, let $D$ be a divisor satisfying the assumptions of Theorem and Corollary, and let $b_D(s)$ be the relevant Bernstein–Sato polynomial. For a well-filt…

Restricted Yano conjecture. The $b$-exponents $\{\tilde{\beta}_1,\ldots,\tilde{\beta}_\mu\}$ of a generic element of $\Sigma_\mu$ satisfy

Arrangement Bernstein–Sato conjecture.

Topological Monodromy Conjecture. Any pole of $Z_f(s)$ yields under exponentiation an eigenvalue of the monodromy operator at some $p\in\operatorname{Var}(f)$. The strong form asse…

Let $f_\mathscr{A}$ define a central reduced indecomposable arrangement of $d$ hyperplanes in $\mathbb{C}^n$. The $n/d$-conjecture. The number $-n/d$ is a root of the Bernstein--Sa…

Let $P(x_1,\ldots,x_n)\not\equiv0$ be a homogeneous polynomial in $n$ variables with coefficients in a field $K$ of characteristic $0$, and let $b(s)$ be its Bernstein–Sato polynom…

Let $k$ be a field of characteristic zero, let $X$ be the relevant $k$-variety, let $f:X\to\mathbb{A}^1_k$ be the morphism, let $x$ be a closed point with residue field $k_x$, and…

Let $f\in\mathbf{C}[x_1,\ldots,x_n]$ be a non-constant polynomial with $f(0)=0$, let $\mu:Y\to\mathbf{C}^n$ be a log resolution of $f^{-1}(0)$, and write … for its local topologica…