  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 $p^\alpha$.
