Christol and, independently, Denef and Lipshitz showed that an algebraic sequence of $p$-adic integers (or integers) is $p$-automatic when reduced modulo $p^\alpha$. Previously, the best known bound on the minimal automaton size for such a sequence was doubly exponential in $\alpha$. We improve this bound to the order of $p^{\alpha^3 h d}$, where $h$ and $d$ are the height and degree of the minimal annihilating polynomial. We achieve this bound by showing that all states in the automaton are naturally represented in a new numeration system. This significantly restricts the set of possible states. Since our approach embeds algebraic sequences as diagonals of rational functions, we also obtain bounds more generally for diagonals of multivariate rational functions.

翻译：Christol 以及 Denef 与 Lipshitz 分别证明了 $p$ 进整数（或整数）的代数序列在模 $p^\alpha$ 约化后是 $p$ 自动的。此前，此类序列的最小自动机规模的最佳已知上界是关于 $\alpha$ 的双重指数级。我们将其改进至 $p^{\alpha^3 h d}$ 量级，其中 $h$ 和 $d$ 分别为最小零化多项式的高度与次数。我们通过证明自动机中的所有状态均可自然地用一种新的记数系统表示来达成这一改进，这极大地限制了可能的状态集合。由于我们的方法将代数序列嵌入为有理函数的对角线，我们也更一般地得到了多元有理函数对角线的界。