AI 中文总结
研究特征 \(p>0\) 域上平面帕斯卡有限自同构，证明其阶由帕斯卡深度决定，阶谱为 \(\{1,p,p^2\}\) 且深度 \(\leq p^2\)。对多项式群给出阶为 \(p^2\) 上限的另一证明，还证明界限精确，对比了平面与高维情况。
AI 中文摘要
设 \(K\) 是特征 \(p>0\) 的域。对于仿射平面的帕斯卡有限自同构 \(F\)，我们证明其阶由帕斯卡深度决定，\(|F| = p^{\lceil\log_p\tau_K(F)\rceil}\)，并且结合多尔加乔夫关于平面克雷莫纳群的定理，这将帕斯卡有限平面自同构的阶谱固定为 \(\{1,p,p^2\}\) 并将帕斯卡深度限制为 \(\tau_K(F)\leq p^2\)。对于多项式群 \(\text{GA}_2(K)\)，我们给出了阶为 \(p^2\) 上限的第二个独立证明，这是一个纯群论论证，使用了容格 - 范德库尔姆融合和塞尔树定理，未使用双有理几何。我们从两个独立意义上证明该界限是精确的。长度为二的维特向量达到阶 \(p^2\)，一个明确的驯自同构 \(G_p\) 达到帕斯卡深度 \(p^2\)，我们给出了一个无特征证明 \(\tau_K(G_p)=p^N\)。我们将平面与更高维度进行对比，在更高维度中阶和深度都是无界的。
英文摘要
Let $K$ be a field of characteristic $p>0$. For a Pascal finite automorphism $F$ of the affine plane we show that its order is determined by its Pascal depth, $|F|=p^{\lceil\log_pτ_K(F)\rceil}$, and that, combined with Dolgachev's theorem on the plane Cremona group, this pins the order spectrum of Pascal finite plane automorphisms to $\{1,p,p^2\}$ and bounds the Pascal depth by $τ_K(F)\le p^2$. For the polynomial group $\text{GA}_2(K)$ we give a second, independent proof of the order-$p^2$ ceiling, a purely group-theoretic argument from the Jung--van der Kulk amalgam and Serre's tree theorem, using no birational geometry. We prove that the bound is sharp in two independent senses. Order~$p^2$ is attained by the length-two Witt vectors, and Pascal depth $p^2$ is attained by an explicit tame automorphism $G_p$, for which we give a characteristic-free proof that $τ_K(G_p)=p^2$. We contrast the plane with higher dimensions, where both order and depth are unbounded.
