27 problems

Let $R$ be a Krull ring, and let $\operatorname{Int}(R)$ denote its ring of integer-valued polynomials. A PVMR is the corresponding Prüfer-v-multiplication ring notion for rings wi…

Let $D$ be a Dedekind domain of characteristic zero with quotient field $K$. The Dedekind-domain containment conjecture. Every idempotent $D$-plethory is contained in $K[e]$, or, e…

Let $D$ be either of the domains … or … Here ${\operatorname{Int}}(D)$ denotes the ring of integer-valued polynomials on $D$, regarded as a $D$-module. Non-flatness conjecture. For…

Integral-closedness criterion. If $\operatorname{Int}_K(A)$ is integrally closed, then $\operatorname{Int}_K(A)$ is Prüfer.

Let ${\mathbb Z}^{+}$ denote the positive integers, and for $n\in{\mathbb Z}^{+}$ define … where … Write $w_n(x)=w_n^{(1)}(x)$, let $(a,b)$ denote the greatest common divisor of…

For $h,m,n\in\mathbb Z^+$, define the generalized Delannoy polynomials … and the generalized Schröder polynomials … where $C_k=\binom{2k}{k}/(k+1)$ and $(a,b,c)$ denotes the greate…

For each integer $n\geq 1$, let $\mathcal{A}_n$ be the set defined in the paper, viewed as a subset of the algebraic integers $\overline{\mathbb{Z}}$. A subset is polynomially clos…

Let $S$ be the set under consideration, let $p$ be a prime, and let $\mathcal{S}_p$ denote the associated $p$-adic set. A sequence in $\mathcal{S}_p$ is pseudo-divergent when it ha…

Z.-W. Sun's stronger congruence conjecture.

Z.-W. Sun's conjecture. The polynomial

Z.-W. Sun's conjecture. The polynomial

Let $n,l\in\mathbb Z^+=\{1,2,3,\ldots\}$ and let $x$ be an indeterminate. Define … A polynomial $P(x)\in\mathbb Q[x]$ is integer-valued when $P(a)\in\mathbb Z$ for every…

Let $\varepsilon\in\{\pm1\}$ and $n,l,m\in\mathbb Z^+=\{1,2,3,\ldots\}$. Define … A polynomial $P(x)\in\mathbb Q[x]$ is integer-valued when $P(a)\in\mathbb Z$ for every…

Let $R_k^B(n)$ denote the number of signed permutations of size $n$ whose burnt-pancake distance from the sorted stack is $k$. Summation identity conjecture. If $k,n\geq1$, then ……

Conjecture. All the maximal ideals of $\operatorname{Int}(E,V)$ lying over $\mathfrak m$ are of the form

Elliott's conjecture. This density exists and is rational, and it is always at least

Elliott's conjecture. The density of the solutions always exists and is rational; moreover, when $x=0$, it is at least $1/d$. This concerns the distribution of characteristic ideal…

Let $p$ and $q$ be nonnegative integers, and let $b_{p,q}$ denote the number of Borel orbits in the corresponding symmetric variety of type $BI$. For each fixed $q$, regard…

Let $S=R=\mathbb{Z}[i]$, and let $(l_1,l_2,\ldots)$ be the sequence of lengths associated with a fixed divisor sequence, where $l_k$ is the smallest number of initial elements dete…

Let $D$ be an integral domain, and let $\operatorname{Int}(D)$ denote its ring of integer-valued polynomials. For a ring $R$, consider the conditions that $\operatorname{Int}(R)$ i…

Let $d_k(x)$ and $s_k(x)$ be the two Delannoy-related polynomials considered in the paper. Integer-valued Delannoy-polynomial power conjecture. For all positive integers $m$ and…

Let $D$ be an integral domain. Non-plethory-domain conjecture. There exists an integral domain $D$ such that ${\operatorname{Int}}(D)$ is not an (idempotent) $D$-plethory. The prec…

S12's conjecture. For every $n\in\mathbb Z^+$,

Asymptotic cardinality conjecture. The minimal cardinality satisfies

Let $n$ and $m$ be positive integers, and let … be the central Delannoy polynomial. Sun's conjecture. … is always an integer. This asserts integrality of the normalized sum of powe…