839 problems

Let $a_{11},a_{22},a_{33},a_{12},a_{13},a_{23},d$ be integers, and consider the generalized Markoff equation … Let $D$ and $A_{11},A_{22},A_{33}$ denote the quantities defined earl…

Prasad and Ram's conjecture. For any $T\in M_{3m}(\mathbb{F}_q)$,

Let $p\geq 3$ be prime. A Hartke $\mathbb{Z}_p$-magic graph is a graph with the Hartke magic-labeling property over $\mathbb{Z}_p$; its order is its number of vertices. Low–Roberts…

Let $\mathbb{K}$ be a finite field of characteristic different from $2$, and let $\mathbb{K}^{*}$ denote its multiplicative group. An element $x$ is a generator if it generates…

Let $p$, $m$, $r$, $s$, and $t$ be positive integers with $p$ prime, let $\boldsymbol{\alpha}=(\alpha_1,\ldots,\alpha_r)\in(\mathbb{F}_{p^m})^r$ have pairwise distinct entries, and…

Let $m=p_1\cdots p_r$ be a square-free product of $r$ distinct primes, and let $K_{n,d}^m$ denote the number of distinct zeros of a random polynomial in…

Let $q$ be an odd prime power and let $n\geq 3$. A quadratic system is a map … where each $f_i\in\mathbb{F}_q[x_1,\dots,x_n]$ has degree at most $2$. Classification conjecture. Suc…

Nonexistence conjecture. The polynomial $f(x)$ is not a permutation polynomial over $\mathbb{F}_{q^2}$.

Classification conjecture. A $\Phi_{\kappa}$-sequence $(a_n)_n$ is complete if and only if

Let $q$ be a prime power, let $\sigma$ be an operator with $\sigma^n=\operatorname{id}$, and let $W_1$ and $W_2$ be $\sigma$-invariant subspaces satisfying … Write $h_{\sigma}(V)$ for…

Let $q=p^m$ and $Q=p^e$. For integers $n$ and $\ell\geq 1$, let $g_{n,q}^{(\ell)}$ be the multivariate polynomial obtained from $g_{n,q}$ by composing with…

Orbit-degree decomposition conjecture. Then $S_n$ splits as a direct sum of fields of degree $|\mathcal O_i|$ for all $i\in\{0,\dots,k\}$.

For every pair of positive integers $n,k$ with $\gcd(k,n)=1$, let $F:\mathbb{F}_{2^n}\to\mathbb{F}_{2^n}$ be defined by $F(x)=x^{4^k-2^k+1}$, and let…

For every prime power $q$ and integer $k\ge 0$, let $C_k(q)$ be the least cardinality of a family of nonzero polynomials over $\mathbb{F}_q$ whose associated codimension-$k$ partia…

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Let $\mathcal{B}_n=\left\{f(x)=\sum_{i=0}^{n}\varepsilon_i x^i:\ \varepsilon_i\in\{0,1\},\ \varepsilon_n=1\right\}$. The Erdős–Mauduit–Sárközy conjecture asserts that there exists…

Gow--McGuire Conjecture 1. Let $q\ne 13$ be any odd prime power. Let $\alpha\in\mathbb{F}_{q^{2}}\setminus\mathbb{F}_{q}$ be fixed. Then there exists $\lambda\in\mathbb{F}_{q}$ suc…

Let $m\ge 8$ be an even integer, let $b,c\in\mathbb{F}_{2^m}$, and let $r$ satisfy $r\le m/2$ and $\gcd(r,m)=1$. Let $f_{b,c,r}$ denote the function defined in Theorem. Non-APN con…

Let $q$ range over powers of $2$, and consider isogeny classes of $3$-dimensional abelian varieties over the finite field of characteristic $2$ with $q$ elements. The classes under…

Budaghyan–Carlet–Helleseth conjecture. No APN function $F \colon \mathbb{F}_2^n \to \mathbb{F}_2^n$ has algebraic degree $n$ for all $n \geq 3$.

Let $S(m)=\mathbb{F}_q[x_0,\dots,x_m]$, and let $S_d(m)$ be its degree-$d$ graded component. For $m,d\geq 1$ and $1\leq r\leq \binom{m+d}{d}$, let $e_r(d,m)$ be the maximum number…

Cyclotomic-number conjecture. If $\ell\geq 3$, then

Ma–Zha–She's quotient-group conjecture. There is a generator $t\langle -1,2\rangle$ of $\mathbb{Z}_p^{\times}/\langle -1,2\rangle$, for some $t\in\mathbb{Z}_p^{\times}$, such that

Let $p$ be an odd prime, let $H=\langle-1,2\rangle\subset\mathbb{F}_p^{\times}$, let $\ell_0$ be the index parameter used for $H$, and let $g$ be a generator of…

Let $\lambda$ be a partition, and let $P(s_\lambda\mapsto0)$ denote the probability that the Schur function $s_\lambda$ evaluates to zero over the finite field under consideration.…