44 problems

Let $P_n(z)$ be the sorted binomial polynomial with parameter $n$. Irreducibility conjecture. $P_n(z)$ is irreducible over $\mathbb{Q}$ for all even $n$. The analogous factorizatio…

Conway's conjecture. The series $1+\sum_n t^{-\frac{1}{n}}$ is irreducible in $\mathbb{Z}+\mathbb{R}((G^{<0}))$. Conway's conjecture was proved by Berarducci, so this statement is…

Let $f\in\mathbb{Z}[x]$ be irreducible and have trivial fixed divisor. Bunyakowski's conjecture. The value $|f(n)|$ should be prime for infinitely many $n\in\mathbb{Z}$. This is a…

Generalized Laguerre Galois-group conjecture. The polynomial $L_n^{(-1-n-r)}(x)$ is irreducible over $\mathbb{Q}$, and its Galois group contains the alternating group $A_n$. For a…

Let $W_{d^2}[n]$ be the locus under consideration, and for odd $n$ let $W_{d^2}^{\epsilon}[n]$, with $\epsilon\in\{0,1\}$, denote its two parity-labelled subloci. Parity conjecture…

Prime-perturbation irreducibility conjecture. The Dirichlet polynomial

Turán-type perturbation conjecture. There is an absolute constant $C>0$ such that, for every such $f$, there is a bounded perturbation $g$ satisfying

Hilbert irreducibility conjecture. The polynomial $f$ remains irreducible for infinitely many integer specializations of $s$.

Let $f:\mathbb Z_{>0}\to\mathbb Z_{>0}$ be a function such that, for every $d$, there is at least one prime with $d<p\leq f(d)$. Choose a prime $p$ uniformly at random from…

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…

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

Let $k$ be a field of characteristic $p\geq 3$, and let $\mathcal{H}_g$ be the moduli space of smooth hyperelliptic curves of genus $g$ over $k$. For $0\leq f\leq g$, write…

Let $k$ be a field of characteristic $p\geq 3$, and let $\mathcal{H}_g$ be the moduli space of smooth hyperelliptic curves of genus $g$ over $k$. Denote by $\mathcal{H}_g^{g-2}$ th…

Let $k$ be a field of characteristic $p\geq 3$, and let $\mathcal{H}_g$ be the moduli space of smooth hyperelliptic curves of genus $g$ over $k$. Write $\mathcal{H}_g^{g-1}$ for it…

Scalarity conjecture. This result should hold for arbitrary $N$: the algebra $\mathcal{D}(W)$ contains only scalar operators of order zero.

Let $K$ be a field and let $b\in K((\mathbb{R}^{\leq 0}))$ be a random generalized power series, meaning that either $\operatorname{cl}(\operatorname{supp}(b))-\{0\}$ is…

Let $f(X)=\sum_{i=0}^n \varepsilon_iX^i$ be sampled uniformly at random from the degree-$n$ Littlewood polynomials, where the coefficients $\varepsilon_i$ are independent and unifo…

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

Stability conjecture. If $f(z)=z^d+\frac{1}{c}$ is irreducible over $\mathbb{Q}$, then $f(z)$ is stable over $\mathbb{Q}$.

Merino's irreducibility conjecture. This polynomial is irreducible over $\mathbb{R}$ if and only if $m$ and $n$ are coprime. The claim gives a precise arithmetic criterion for irre…

For $d,K\in\mathbb{N}$, define … Choose the coefficients independently and uniformly from the permitted ranges. Growing-range irreducibility conjecture. If $f\in\mathcal{P}_{d,K}$,…

Asymptotic Reducibility Conjecture. The proportion of polynomials in $\mathcal{B}=\bigcup_{N\geq 2}\mathcal{B}^{(N)}$ which are irreducible is given by a limit, assumed to exist, e…

Irreducibility conjecture. If $|a|\geq 5$, then $x^{n}+ax^{m}\pm1$ is irreducible over $\mathbb{Q}$. Furthermore, there are only finitely many reducible polynomials of the form…

Let $q$ be a prime power, and let $\mathcal{F}_1,\dots,\mathcal{F}_r\in\mathbb{F}_q[T][X]$ be irreducible and separable in $X$. Function-field Schinzel hypothesis H. If…

Irreducibility conjecture. The polynomial $R_n(x;\ell_1,\ell_2,y)$ is irreducible unless $\ell_1=\ell_2$ and $n$ is odd. In that exceptional case, $R_n(x;\ell_1,\ell_2,y)$ is the p…