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Are there any restrictions on the ground field over which the implicit function theorem holds? In particular, does the theorem hold over function fields like $F_q((1/t))$?

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The field $\mathbb{F}_q((1/t))$ is not a "function field", at least not according to my philosophy. Perhaps you could specify what you mean by a function field and also what statement counts as "the implicit function theorem" over your function fields? –  Pete L. Clark Mar 30 '11 at 17:22
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There is an implicit function theorem valid for any non-Archimedean field. See Theorem 2.2.1 in J.-I. Igusa, An introduction to the theory of local zeta functions. AMS, 2000.

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