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This question raised another question in my mind. In finite fields $F$, every function from $F$ to itself is a polynomial. What about infinite fields of finite (i.e. non-zero) characteristic? Are there non-polynomial functions there?

---- Naively

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    Sure, non-polynomial is any function $\ne 0$ with infinitely many roots, e.g. delta functions, step functions, etc.2012-03-07

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