Square root of a positive $C^\infty$ function. - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-23T19:56:43Z http://mathoverflow.net/feeds/question/105438 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/105438/square-root-of-a-positive-c-infty-function Square root of a positive $C^\infty$ function. Michael Barr 2012-08-25T02:25:35Z 2012-08-25T04:40:21Z <p>Suppose $f$ is a $C^\infty$ function from the reals to the reals that is never negative. Does it have a $C^\infty$ square root? Clearly the only problem points are those at which $f$ vanishes. </p> http://mathoverflow.net/questions/105438/square-root-of-a-positive-c-infty-function/105442#105442 Answer by Dick Palais for Square root of a positive $C^\infty$ function. Dick Palais 2012-08-25T03:47:26Z 2012-08-25T04:40:21Z <p>The answer is "no". This is covered in great detail here:</p> <p><a href="http://www.math.polytechnique.fr/~bony/BBCP_jfa.pdf" rel="nofollow">http://www.math.polytechnique.fr/~bony/BBCP_jfa.pdf</a></p> http://mathoverflow.net/questions/105438/square-root-of-a-positive-c-infty-function/105445#105445 Answer by Evan Jenkins for Square root of a positive $C^\infty$ function. Evan Jenkins 2012-08-25T04:10:20Z 2012-08-25T04:10:20Z <p>The function <code>$$f(x) = \begin{cases} \sin^2 \left(\frac{1}{x} \right) e^{-1/x} + e^{-2/x} &amp; \text{if x &gt; 0,}\\ 0 &amp; \text{if x \leq 0,} \end{cases}$$</code> is $C^\infty$ but has no $C^2$ square root. I found this example in the paper <a href="http://www.ams.org/mathscinet-getitem?mr=1639759" rel="nofollow">Choosing roots of polynomials smoothly</a> by Alekseevsky, Kriegl, Losik, and Michor (available freely <a href="http://www.mat.univie.ac.at/~michor/roots.pdf" rel="nofollow">here</a>). This example appears to have come from Frank Warner's (unpublished) 1963 dissertation. </p>