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Does this argument suppose non-degeneracy? The claim seems to hold also in the degenerate case, but "each of the operators $D−I ω_n$ has a one-dimensional kernel" seems to require that each root is simple...
The degree of non-constructivity of Brouwer's fixed point theorem in various dimensions is explored in the following publication: link.springer.com/chapter/10.1007%2F978-3-642-30870-3_7 See in particular reference [12] to the paper by Orevkov in 1963!
