4 problems

Hutz's conjecture. There is no even degree $d>2$ and no $c\in\mathbb Q$ such that $\varphi_{d,c}$ has a rational point of exact period $n$. Moreover,

Milnor's Galois-orbit conjecture. These values of $c$ form one Galois orbit under the action of the absolute Galois group of $\mathbb{Q}$.

Let $k$ be a number field, let $S$ be a finite set of places of $k$ including all the archimedean places, and let $d\geq 2$ be an integer. Write $f_{d,c}(z):=z^d+c$. Let $D$ be a n…

For an integer $d\geq 2$ and $c\in\mathbb{Q}$, let $f_{d,c}(z)=z^d+c$. Generalized Poonen conjecture. For $n>3$, there is no $f_{d,c}$ defined over $\mathbb{Q}$ with a $\mathbb{Q}$…