Fix $d \ge 2$ and a field $k$ such that $\mathrm{char}~k \nmid d$. Assume that $k$ contains the $d$th roots of $1$. Then the irreducible components of the curves over $k$ parameterizing preperiodic points of polynomials of the form $z^d+c$ are geometrically irreducible and have gonality tending to $\infty$. This implies the function field analogue of the strong uniform boundedness conjecture for preperiodic points of $z^d+c$. It also has consequences over number fields: it implies strong uniform boundedness for preperiodic points of bounded eventual period, which in turn reduces the full conjecture for preperiodic points to the conjecture for periodic points.

中文翻译：

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动力曲线的有向性和前周期点的强均匀有界性

修正 $d \ge 2$ 和一个字段 $k$ 使得 $\mathrm{char}~k \nmid d$。假设 $k$ 包含 $1$ 的第 $d$ 个根。然后，在参数化形式为 $z^d+c$ 的多项式的前周期点的 $k$ 上的曲线的不可约分量在几何上是不可约的，并且具有趋向于 $\infty$ 的共性。这意味着 $z^d+c$ 的前周期点的强均匀有界猜想的函数场模拟。它也对数域产生影响：它意味着有界最终周期的前周期点的强一致有界性，这反过来又将前周期点的完全猜想简化为周期点的猜想。