论文标题
某些高度多项式的稳定性
Stability of Certain Higher Degree Polynomials
论文作者
论文摘要
算术动力学中有趣的问题之一是研究多项式在一个领域的稳定性。在本文中，我们研究了$ f（z）= z^d+\ frac {1} {c} $ $ d \ geq 2 $，$ c \ in {\ mathbb {z} \ setMinus \ {0 \}} $的稳定性。我们表明，对于$ d \ geq 3 $的无限家庭，每当$ f（z）$都是不可修复的，其所有迭代都是不可约的，也就是说，$ f（z）$都是稳定的。对于$ c \ equiv 1 \ pmod {4} $，我们表明$ z^2+\ frac {1} {c} {c} $的所有迭代都是不可记录的。另外，我们还表明，对于$ d = 3 $，如果$ f（z）$可还原，则$ f（z）$的每个迭代的不可约合因子的数量正好为$ 2 $，对于$ | c | c | \ leq {10^{12}}} $。
One of the interesting problems in arithmetic dynamics is to study the stability of polynomials over a field. In this paper, we study the stability of $f(z)=z^d+\frac{1}{c}$ for $d\geq 2$, $c\in{\mathbb{Z}\setminus\{0\}}$. We show that for infinite families of $d\geq 3$, whenever $f(z)$ is irreducible, all its iterates are irreducible, that is, $f(z)$ is stable. For $c\equiv 1\pmod{4}$, we show that all the iterates of $z^2+\frac{1}{c}$ are irreducible. Also we show that for $d=3$, if $f(z)$ is reducible, then the number of irreducible factors of each iterate of $f(z)$ is exactly $2$ for $|c|\leq{10^{12}}$.
