Christol's theorem states that a power series with coefficients in a finite field is algebraic if and only if its coefficient sequence is automatic. A natural question is how the size of a polynomial describing such a sequence relates to the size of an automaton describing the same sequence. Bridy used tools from algebraic geometry to bound the size of the minimal automaton for a sequence, given its minimal polynomial. We produce a new proof of Bridy's bound by embedding algebraic sequences as diagonals of rational functions. Crucially for our interests, our approach can be adapted to work not just over a finite field but over the integers modulo $p^\alpha$.

翻译：Christol 定理指出，系数在有限域中的幂级数是代数的当且仅当其系数序列是自动的。一个自然的问题是，描述该序列的多项式的大小与描述同一序列的自动机的大小之间存在何种关系。Bridy 利用代数几何工具，在给定序列的最小多项式的情况下，给出了其最小自动机大小的上界。我们通过将代数序列嵌入有理函数的对角线上，为 Bridy 的上界提供了一种新的证明方法。关键在于，我们的方法不仅适用于有限域，还可推广到模 $p^\alpha$ 的整数环上。