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Are extensions of profinite groups profinite?

If a formal power series over the complex numbers satisfies a polynomial identity, does it imply that the power series has a radius of convergence?

two non-degenerate quadratic forms on $GF(2)^2r$

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quadratic-forms
profinite-groups
power-series
matrices
graph-theory
gr.group-theory
cv.complex-variables
complex-analysis
ac.commutative-algebra

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