Timeline for If $f$ is smooth and even then there exists a smooth function $g$ such that $f(x)=g(x^2)$
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| Jul 26, 2021 at 13:01 | comment | added | Balarka Sen | @TimCampion Whitney contributed much to structures of germs of smooth functions. This fact is saying that a Z/2-invariant germ (R, 0) -> (R, 0) lifts along x^2 -- this is a consequence of Malgrange preparation theorem, which among many other things, proves that germ of a generic smooth mapping (R^2, 0) -> (R^2, 0) is equivalent to either identity, (x^2, y) (fold) or (x^3 + xy, y) (cusp), a theorem also originally by Whitney. These questions gained relevance only after people started thinking systematically about singularities rather than smoothness. | |
| Aug 24, 2020 at 20:58 | comment | added | Tim Campion | Wow -- I wouldn't have been surprised if the theorem were 100 years older than this! It would seem the necessary tools were available earlier... Perhaps it was necessary for people to start thinking systematically about differential topology for such a question to attain relevance? | |
| Aug 9, 2011 at 18:36 | history | answered | Igor Rivin | CC BY-SA 3.0 |