71 problems

Let $\mathbb{K}$ be a field of characteristic $0$, and let $f(X)$ be a monic univariate polynomial of degree $d$. Write $f_i$ for the $i$-th derivative of $f$. Assume that…

Let $p,q$ be coprime positive integers with $p\neq q$, and let $Q_{p,q}(t)\in\mathbb{Z}[t]$ be Sharipov's second cuboid polynomial of degree $10$. Sharipov's irreducibility conject…

Gall-Rahav conjecture. The polynomial $\sigma(M^{2h})$ is divisible by a non-Mersenne prime.

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

Let $q>2$ be an integer, and let $r_n$ denote the probability that a random monic degree-$n$ $p$-adic polynomial splits into linear factors. Let $\bar W_q$ be a continuous function…

Higher-arity recursion conjecture. The same recursion should hold when $q>2$ is integral:

Stable-factorization conjecture. The polynomial $f_r$ is eventually stable over $\mathbb{Q}(i)$ with constant $C(f_r,0)=4$, and exactly one of the following cases holds:

For binary partitions, let $\operatorname{num}_{\mathcal B}(n,x)$ and $\operatorname{den}_{\mathcal B}(n,x)$ denote the corresponding numerator and denominator, where binary parts…

Let $\operatorname{num}(n,x)$ and $\operatorname{den}(n,x)$ be the ordinary numerator and denominator of the reciprocal of the partition polynomial for $n\geq 1$. Ballantine–Beck–F…

Let $\operatorname{num}(n,x)$ denote the ordinary partition numerator for $n\geq 1$. Ballantine–Beck–Feigon–Maurischat's irreducibility conjecture. For every $n\geq 1$, the polynom…

For a prime $p$, define the Fekete polynomial … where $\Big(\frac{a}{p}\Big)$ is the Legendre symbol. MNT23's conjecture. For every prime $p$, $F_p(x)$ is a product of linear facto…

Difference-multiset injectivity conjecture. As $n \to \infty$, the image of $\phi$ contains $2^{n-1}+o(2^{n})$ distinct elements. For a set $A$ with maximum element $a'$, the refle…

Double-discriminant factor irreducibility conjecture. The polynomial $B_{n,0}$ is irreducible; indeed,

Double-discriminant factorization conjecture. For $0\leq k\leq n/2$, there is a factorization in $\mathbb{Z}[a_0,\ldots,\widehat{a_k},\ldots,a_n]$,

Rank-factorization conjecture. For any partition $\lambda$ of rank $r$, there is a polynomial $Q_\lambda(z;q,t)\in\mathbb{Z}_{\geq 0}[z,q,t]$ such that

Let $K$ be a field, let $n$ be a positive integer, and set $R=K[x_1,\ldots,x_{n-1}]$. For $j\in\{1,\ldots,n-1\}$, let $\Phi_j:R\to R$ be given by $\Phi_j(x_i)=x_i-x_j$ for…

Gimbert's cyclotomic irreducibility conjecture. If $k$ is even, then $F_{i,k}(x)$ is reducible in $\mathbb Q[x]$ if and only if $i\mid(k+2)$, and in that case it has exactly two fa…

Goh–Wildberger conjecture. There are polynomials $\Phi_d(x)\in\mathbb{Z}[x]$, $d\geq 1$, such that, for every $n\geq 1$,

Sa conjecture. The set of polynomials over finite fields that do not satisfy any of the conditions in Sa is empty.

Let $q$ be an odd prime, and let $R_q(y)$ be the resultant polynomial defined in the paper. Norm-form conjecture. There exist polynomials $h_1,h_2\in\mathbb Z[y]$ such that … The p…

Define … Let $\Phi_r(q)$ denote the cyclotomic polynomial of order $r$. The normalized cusp-polynomial irreducibility conjecture. The polynomial $P_d(t;q)$ lies in…

Let $\nu\geqslant3$, let $\tilde f_0,\tilde f_1,\ldots,\tilde f_{\nu-1}$ be algebraically independent variables, and let $F_\nu(z)$ be the matrix introduced in the paper. Regard…

Irreducibility conjecture. If $p\nmid a_{n}a_{q}$, $q\nmid u$, and

Let $h(z)$ be a degree-$n$ polynomial over a local field with residue-field size $q$, and let $\rho(n,\sigma;q)$ denote the density of polynomials whose associated finite étale alg…

Generalized density functional-equation conjecture. The density satisfies