45 problems

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…

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 $K$ be a field, and let $f\in K[x]$ be a non-constant monic univariate polynomial of degree $n$. For $1\leq i\leq n-1$, write $H_i(f)$ for its $i$-th Hasse derivative, and call…

Let $f(z)=z^3+\frac{1}{c}$ with $c\in\mathbb{Z}\setminus\{0\}$. Here $f^n$ denotes the $n$-fold iterate of $f$. Eventual stability conjecture. If $f(z)$ is reducible over…

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…

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…

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 $n\geq 1$,

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…

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…

Let $n$ be a positive integer and let $\sigma$ be a splitting type of degree $n$. For a prime $p$, let $\rho(n,\sigma;p)$ be the Haar measure of the $p$-adic polynomials of degree…

Let $X,Y$ be discrete independent random variables with finite support in the natural numbers, and suppose that $Z=X+Y$ is uniform on its support. Equivalently, let $R(t)=P(t)Q(t)$…

Boston–Jones exceptional-orbit conjecture. The distribution of the factorization process of $f$ converges to limiting distributions arising from the Boston–Jones model unless $f$ h…

Let $K$ be a number field, let $\phi\in K(x)$, and let $\alpha\in K$. The pair $(\phi,\alpha)$ is eventually stable when, writing $\phi^n=p_n/q_n$ with coprime $p_n,q_n\in K[x]$, t…

Carnevale–Voll's factorisation conjecture. The polynomial giving the joint distribution of $(\operatorname{den},\operatorname{exc})$ over $S_{\eta}$ has a unitary factor if and onl…

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…

Inheritance-property conjecture. All hyperfields with the doubly distributive property satisfy the inheritance property.

Multiplicity-bound conjecture. All hyperfields with the doubly distributive property satisfy the multiplicity bound.