The author is grateful to Lojasiewicz and Glaeser for helpful discussions. In particular Glaeser pointed out the possibility of linearizing Nirenberg's lemma.
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References
L. Nirenberg, A new proof of the preparation theorem. This volume, pp.97–105.
J. N. Mather, Stability of C∞ mappings: I. The division theorem. Annals of Math., 87 (1968) pp. 89–104.
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Mather, J.N. (1971). On nirenberg's proof of malgrange's preparation theorem. In: Wall, C. (eds) Proceedings of Liverpool Singularities — Symposium I. Lecture Notes in Mathematics, vol 192. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0066816
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DOI: https://doi.org/10.1007/BFb0066816
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