13 problems

Let $K$ be a Hilbertian field of characteristic $0$, and let $n\geq 2$. For a polynomial $f\in K[x]$, write $f^{-k}(0)$ for the set of $k$-th preimages of $0$, and let $[S_n]^k$ de…

Let $K$ be a number field, let $K^{\mathrm{ab}}$ be its maximal abelian extension, let $f:{\mathbb P}^1\to{\mathbb P}^1$ be a polynomial map of degree $d\geq2$ defined over $K$, an…

Let $K$ be a global field of characteristic not $2$, let $f\in K[x]$ be a quadratic polynomial, and let $\alpha\in K$. Suppose that the backward orbit … contains no ramification po…

Let $K$ be the field from which the rational map is defined, let $\phi\in K(x)$ be a rational function of degree $2$, and let $\mathcal{T}$ be the complete rooted $2$-ary tree atta…

Hyperbolic polynomial branch and extension conjecture. The following are equivalent:

Let $K$ be a number field, let $f:{\mathbb P}^1\to{\mathbb P}^1$ be a rational map of degree at least $2$ defined over $K$, and let $\alpha\in{\mathbb P}^1(K)$ be non-exceptional f…

Let $f:{\mathbb P}^1\to{\mathbb P}^1$ be a rational map of degree at least $2$. A power map is a rational map $\mathrm{PGL}_2(\bar K)$-conjugate to $x^{\pm d}$; a Chebyshev map is…

Let $K$ be a number field, let $f:{\mathbb P}^1\to{\mathbb P}^1$ be a rational map of degree $d\geq2$ defined over $K$, and let $\alpha\in{\mathbb P}^1(K)$. Minkowski-dimension exi…

Orbital open image conjecture. The representation has open image, equivalently, finite index. This is a dynamical analogue of the Néron–Ogg–Shafarevich perspective on arboreal Galo…

Let $k$ be a field of characteristic different from $2$, and let $t$ be transcendental over $k$. For the polynomial … let its arboreal representation be the natural Galois action o…

Let $K$ be a global field of characteristic $0$ or greater than $2$, and let $\phi(x)\in K(x)$ have degree $2$. Assume that $\phi$ is not post-critically finite and that $0$ is not…

Let $\phi(x)=b(x^2+1)/x$ with $b\in\mathbb{Q}$ and $b\notin\{0,\pm 1/2\}$. Let $T_\infty$ be the pre-image tree rooted at $0$, let $G_\infty(\phi)$ be its arboreal Galois group, an…

Let $K$ be a field, let $\phi\in K(x)$, and let $T_\infty$ be its pre-image tree rooted at $0$. Let $A(\phi)$ be the subgroup of $\operatorname{Aut}(T_\infty)$ generated by the act…