17 problems

Let $f$ be a polynomial over $\mathbb C$, and let $f_\mu$ denote its reduction to characteristic $p$ for a maximal ideal $\mu$ of a finitely generated subring of $\mathbb C$ contai…

Blickle–Mustaţă–Smith's equality conjecture.

Let $(R,\mathfrak{m})$ be a $d$-dimensional F-finite F-pure non-$\mathbb{Q}$-Gorenstein normal local ring of characteristic $p>0$. Let $\mathrm{c}(\mathfrak{m})$ denote its F-pure…

Denominator conjecture. If $p\equiv-c\pmod{\operatorname{lcm}(a,b)}$, then the denominator of $\operatorname{fpt}(f)$ is $p^e$ for some positive integer $e\leq c$.

Threshold-equality conjecture. There are infinitely many primes $p$ for which

Threshold-equality conjecture. There are infinitely many primes $p$ such that

Let $k$ be a field of characteristic $p$, and let $d\geq 4$ be an integer not divisible by $p$. Let $e$ be the smallest positive integer such that … For a rational number, write…

Blickle–Mustaţă–Smith's accumulation-point conjecture. If $d>1$, then every accumulation point of $\mathcal{T}_{d,p}$ belongs to $\mathcal{T}_{d-1,p}$.

Infinitude conjecture. The set $\mathcal{P}$ has infinite cardinality.

ACC conjecture. The sequence is eventually constant:

Let $f\in\mathbb{Z}[x_0,\ldots,x_N]$, let $f_p\in\mathbb{F}_p[x_0,\ldots,x_N]$ be its reduction modulo $p$, and write $\operatorname{fpt}(f_p)$ and $\operatorname{lct}(f)$ for the…

Infinitely many primes conjecture. The equality

Elliptic-prime conjecture. For any polynomial $f_{cmathbb{Q}}$ over $cmathbb{Q}$, there exist infinitely many primes $p$ such that

Let $f$ be a homogeneous polynomial with rational coefficients, and for each prime $p$ let $f_p$ denote its corresponding model in characteristic $p$. Write $\mathrm{fpt}(f_p)$ for…

Let $A$ be the localization of ${\mathbb{Z}}$ at a nonzero integer, let ${\mathfrak{a}}$ be a nonzero ideal of $A[x_1,\dots,x_n]$ contained in $(x_1,\dots,x_n)$, and let…

Polynomial-realization conjecture. For every prime $p$ and every $n$, every $F$-pure threshold in dimension at most $n$ can be obtained as the $F$-pure threshold at the origin of s…

ACC conjecture. For every prime $p$ and every $n$, the set ${\mathcal T}_n$ satisfies the Ascending Chain Condition, i.e. it contains no strictly increasing sequences.