14 problems

Let $f\in\mathbb{Q}[x]$ be a polynomial, and for $a\in\mathbb{Q}$ consider the iterates $f^{\circ n}(x)-a$. The polynomial $f$ is eventually stable over $a$ when the number of irre…

Let $f$ be a polynomial over a global field $K$. For $alpha$ in $K$ and $N$ in $\mathbb{Z}_{\geq 0}$, let $K_N(f,\alpha)=K(f^{-N}(\alpha))$ be the arboreal field and let…

Let $K$ be a finite extension of $\mathbb{Q}$ or $\mathbb{F}_p(t)$, where $p$ is prime. For integers $d\geq 2$, define $K\in\mathcal{N}_{d,n}^{\infty}$ to mean that $K$ admits the…

For a field $K$, let $K\in\mathcal{N}_{d,n}^{\infty}$ mean that there exists a degree-$d$ polynomial over $K$ whose first $n$ iterates, and all further iterates, remain irreducible…

Nonsquareness conjecture. For every $n\geq3$, $b_n(c)$ is not a square in $\mathbb{Q}$. By the cited lemma, this would imply the irreducibility persistence conjecture and hence eve…

Let $f_r(x)=x^2+r$ with $r=1/c$ for $c\in\mathbb{Z}\setminus\\{0,-1\\}$. Write $f_r^n$ for the $n$-fold iterate. Irreducibility persistence conjecture. If $f_r^2(x)$ is irreducible…

Let $f_r(x)=x^2+r$ with $r=1/c$ for $c\in\mathbb{Z}\setminus\\{0,-1\\}$. Let $C(f_r,0)$ denote the eventual number of irreducible factors of the iterates, and let $k_n$ be the numb…

Let $p$ be a prime, let ${\mathbb F}_p$ be the finite field with $p$ elements, and let $f(x)\in{\mathbb F}_p[x]$ be a polynomial whose degree is divisible by $p$. A polynomial over…

Let $K$ be a field of characteristic different from $2$, and let $f(x)\in K[x]$ be a monic, post-critically finite quadratic polynomial. Let $o_f$ denote the period of the post-cri…

Let $S$ be the set given in Proposition 3.8. For $m\in S$, define … Stability conjecture. For every $m\in S$, the polynomial $f$ is stable, meaning that every iterate $f^n$ is irre…

Uniform dynamical Mordell--Lang conjecture. The set of pairs $(n,m)$ such that

Bound conjecture for $K=3$.

Let $\overline{f}$ denote the reduction of a polynomial $f$ modulo a prime. Odoni–Stoll conjecture. For … $\overline{f}$ is stable for $p=3$ and for no other primes. Odoni observed…

Let $F$ be a global field, and let $f\in F[x]$ be monic and quadratic with critical point $\gamma$. Define … Here, the affine span of $S$ is understood as a subset of $F^*/F^{*2}$.…