论文标题
每个程度的多项式家庭都没有理性的前观点
Families of polynomials of every degree with no rational preperiodic points
论文作者
论文摘要
令$ k $为一个数字字段。鉴于k [x] $ of度量$ d \ ge 2 $的多项式$ f（x）\，可以推测$ f $的预处理点的数量仅取决于$ d $ and $ d $和$ [k：\ mathbb q] $的均匀界限。但是，当$ d $可除以$ 2 $或$ 3 $，$ k = \ m缩Q $时，唯一的多项式参数家族的唯一示例是没有预碘的。在本文中，如果有任何整数$ d \ ge 2 $，我们显示了$ f_t（x）= x^d+c（t）$，$ c（t）\ in K（t）$中的多项式的许多参数家族，没有任何合理的预科点。
Let $K$ be a number field. Given a polynomial $f(x)\in K[x]$ of degree $d\ge 2$, it is conjectured that the number of preperiodic points of $f$ is bounded by a uniform bound that depends only on $d$ and $[K:\mathbb Q]$. However, the only examples of parametric families of polynomials with no preperiodic points are known when $d$ is divisible by either $2$ or $3$ and $K=\mathbb Q$. In this article, given any integer $d\ge 2$, we display infinitely many parametric families of polynomials of the form $f_t(x)=x^d+c(t)$, $c(t)\in K(t)$, with no rational preperiodic points for any $t\in K$.
