I am reading up on Christol's theoreom and an important part is that k-uniform transducers (where k is somehow related to prime numbers) preserve the algebricity of a formal power series (taking the coefficients as a sequence). So basically, we start with a formal power series that has roots (my friend told me that algebraic means there are roots or zero crossings), run the coefficients through a k-uniform transducer and, when we use the values that come out as coefficients for a power series, that power series is again algebraic.
There is a lot I still need to understand here clearly (like, what is a "formal" power series vs a "business casual" power series...), but I'd like to start by understanding what is so special about a power series having roots.
Just for context, I have nearly no formal background in mathematics but came across Christol's theorem and thought it was very cool and maybe related to my area of work (computer science, temporal logics, and automata). Any help is appreciated!