论文标题
单位方程在数量字段中没有质量数字的解决方案至$ 3 $，其中$ 3 $完全拆分
The unit equation has no solutions in number fields of degree prime to $3$ where $3$ splits completely
论文作者
论文摘要
令$ k $为一个数字字段，带有整数$ \ Mathcal O_ {K} $。我们证明，如果$ 3 $不划分$ [k：\ mathbb q] $，而$ 3 $则完全分为$ k $，则单位方程在$ k $中没有解决方案。换句话说，没有$ x，y \ in \ mathcal o_ {k}^{\ times} $，带有$ x + y = 1 $。我们的基本$ p $ -adic证明是受Skolem-chabauty-coleman方法的启发，该方法应用于投影线的标量限制，减去三分。将此结果应用于算术动力学中的问题，我们表明，如果$ f \ in \ mathcal o_ {k} [x] [x] $在$ \ mathcal o_ {k} $中具有有限的环状轨道，则长度为$ n $ n $，然后$ n \ in \ in \ in \ n \ in \ {1，2，4 \} $。
Let $K$ be a number field with ring of integers $\mathcal O_{K}$. We prove that if $3$ does not divide $ [K:\mathbb Q]$ and $3$ splits completely in $K$, then the unit equation has no solutions in $K$. In other words, there are no $x, y \in \mathcal O_{K}^{\times}$ with $x + y = 1$. Our elementary $p$-adic proof is inspired by the Skolem-Chabauty-Coleman method applied to the restriction of scalars of the projective line minus three points. Applying this result to a problem in arithmetic dynamics, we show that if $f \in \mathcal O_{K}[x]$ has a finite cyclic orbit in $\mathcal O_{K}$ of length $n$ then $n \in \{1, 2, 4\}$.
