Gonality of dynatomic curves and strong uniform boundedness of preperiodic points
Gonality of dynatomic curves and strong uniform boundedness of preperiodic points
复制标题
动态曲线的调性和前周期点的强均匀有界性
作者：
J. Doyle;B. Poonen
修复$d\geqslant 2$和一个字段$k$，使$\operatorname{char}k\nmid d$。假设$k$包含$1$的$d$次根。然后在$k$上参数化形式为$z^{d}+c$的多项式的周期前点的曲线的不可约分量是几何上不可约的，并且具有趋向于$\infty$的共向性。这暗示了$z^{d}+c$的前周期点的强一致有界猜想的函数场模拟。它在数域上也有结果：它暗示了最终周期有界的预周期点的强一致有界性，从而将预周期点的完全猜想简化为周期点的猜想。我们的证明包括一个特定于有限域的新颖论证，以及更多的标准工具，如Castelnuovo-Severi不等式。
Fix $d\geqslant 2$ and a field $k$ such that $\operatorname{char}k\nmid d$. 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. Our proofs involve a novel argument specific to finite fields, in addition to more standard tools such as the Castelnuovo–Severi inequality.
影响因子：
0.9
作者：
DOYLE, JOHN R.;KRIEGER, HOLLY;OBUS, ANDREW;PRIES, RACHEL;RUBINSTEIN-SALZEDO, SIMON;WEST, LLOYD
通讯作者：
WEST, LLOYD
动力学一致有界性和 abc 猜想
影响因子：
3.1
作者：
Looper, Nicole R.
通讯作者：
Looper, Nicole R.