44 problems

Let $R$ be a ring with property (T), characteristic $p$, and order $p^{nr}$, whose maximal Galois subfield is $F={\bf F}_p$. Fix the invariants $p$, $n$, $r$, $s$, $t$, and…

Typical finite ring conjecture. The fraction of isomorphism classes of commutative rings $A$ of order at most $N$ satisfying all of the following conditions tends to $1$:

Let $p>6$ be a prime, let $Z$ denote the residue ring modulo $p^k$, and let $F$ be the set of $p$-th power residues in the multiplicative semigroup of $Z$. For $i\geq 1$, write…

Let $R$ be a finite non-commutative CC-ring, meaning that all centralizers of non-central elements of $R$ are commutative, and let $\Gamma_R$ be its commuting graph. Let MSN-integr…

Let $R$ be a finite CC-ring, meaning that the centralizer of every non-central element of $R$ is commutative, and let $\Gamma_R$ be its commuting graph. A graph is CN-hyperenergeti…

Let $p_n$ denote the $n$th prime, and define … Let $\mathcal{C}_{\mathcal{A}}$ denote the subring of algebraic elements of $\mathcal{A}$. Transcendence conjecture. … The paper prov…

Power-weight duality conjecture. The power weights $w^{(\ell)}$, $\ell \geq 2$, do not respect duality over any $\mathbb{Z}/m\mathbb{Z}$ with $m \geq 4$.

Kuijper–Pinto's conjecture. Every convolutional code over $\mathbb{Z}_{p^r}$ admits a minimal $p$-encoder.

Let $n$ be a natural number, and let $G(\mathbb{Z}_n)$ be the unit graph. Write $U(\mathbb{Z}_n)$ for its set of units, $N_U(\mathbb{Z}_n)$ for its set of non-units, and…

Let $p$ be a prime, and let $g_s(p)$ denote the least stationary primitive root in $\mathbb{Z}/p\mathbb{Z}$. Let $\varepsilon>0$ be a small number. Least stationary primitive root…

Let $A$ be a nonzero commutative unitary ring, let $R\subseteq A$, and let … A solution $(a_1,\ldots,a_n)$ of $M_n(a_1,\ldots,a_n)=\pm Id$ is an irreducible solution if it is an ir…

Let $p>3$ be a prime, let $n_1,n_2>1$, and let $E_n(1,x)$ denote the reversed Dickson polynomial of the second kind over $\mathbb{Z}_p$. Reversed Dickson second-kind period conject…

Let $\mathfrak{g}$ be a nilpotent $\mathbb{Z}$-Lie algebra finitely generated as an abelian group. For a finite truncated valuation ring $R$ with maximal ideal $\mathfrak{m}$, writ…

Let $H^{t_1,ots,t_s}$ be a $Z{3^s}$-linear generalized Hadamard code of length $3^t$ with $t1teq 2$ and $ts=0$. Suppose that $CH(t1,ots,ts)$ is the chain of equivalences associate…

Let $f$ be a polynomial in $n$ variables, with $s$ and $d$ as in the preceding setup, and define … Here $N\geq 1$ and $\psi:\mathbb Z/N\mathbb Z\to\mathbb C^\times$ is a group mono…

Let $N,n\in\mathbb{N}$, and let $S\subseteq (\mathbb{Z}/N\mathbb{Z})^n$ be a Kakeya set, meaning that $S$ contains a line in every direction in…

Let $R=\mathbb{Z}_{n}$, where … with primes $p_{1}<p_{2}<p_{3}<\cdots<p_{k}$, including $2$. Write $\psi(n)$ for the quantity appearing in the conjecture. Maximal-index conjecture.…

Let $\mathfrak R$ be the class of finite rings, and let $\mathfrak{R}_{sa}$ denote the class of finite semiassociative rings. Write $\mathfrak{S}_c(\mathfrak R)$ for the set of com…

Middle-weight conjecture. Then $w_2=n$.

Conjecture 9.4. The largest Parker ring of the form $\mathbb{Z}/n\mathbb{Z}$ is

Conjecture 9.3. Among Parker rings of the form $\mathbb{Z}/n\mathbb{Z}$ with $n$ odd, the largest is

Conjecture 9.2. If $\mathbb{Z}/n\mathbb{Z}$ is a record modulus, then

Soit $A$ un $p$-anneau fini, c'est-à-dire un anneau fini dont la caractéristique est une puissance de $p$, et soit $K$ un corps tel que $\mathrm{car}(K)\neq p$. Pour un foncteur…

For integers $n\geq 2$ and $d\geq 1$, define the unitary finite Euclidean graph $T_n^{(d)}$ by … Here $\mathbb{Z}_n^{\times}$ is the set of units of $\mathbb{Z}_n$. Li–Vinh's conje…

Let $n>1$ be an integer, and let $R=\mathbb{Z}_n$ be the ring of integers modulo $n$. The modular Erdős–Burgess constant conjecture. … This conjecture gives an exact formula for th…