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I'm looking forward to know about applications of Christol theorem mentioned in Jefrrey Shallit's Number theory and formal languages. One of them is purely algebraic: if $f, g \in \mathbb{F}_q[[z]]$ are algberaic then their Hadamard product $f \odot g$ is also algebraic.

Can you name other applications?

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There are lots of applications to transcendence in finite characteristic. Christol's theorem makes it possible to give proofs of theorems about transcendence of formal power series using the tools of formal language theory. This was pioneered by Allouche, Valerie Berthe in her thesis, etc.

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