On square roots of class ${C}^{m}$ of nonnegative functions of one variable
Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Série 5, Volume 9 (2010) no. 3, p. 635-644
We investigate the regularity of functions $g$ such that ${g}^{2}=f$, where $f$ is a given nonnegative function of one variable. Assuming that $f$ is of class ${C}^{2m}$ ($m>1$) and vanishes together with its derivatives up to order $2m-4$ at all its local minimum points, one can find a $g$ of class ${C}^{m}$. Under the same assumption on the minimum points, if $f$ is of class ${C}^{2m+2}$ then $g$ can be chosen such that it admits a derivative of order $m+1$ everywhere. Counterexamples show that these results are sharp.
Classification:  26A15,  26A27
@article{ASNSP_2010_5_9_3_635_0,
author = {Bony, Jean-Michel and Colombini, Ferruccio and Pernazza, Ludovico},
title = {On square roots of class $C^m$ of nonnegative functions of one variable},
journal = {Annali della Scuola Normale Superiore di Pisa - Classe di Scienze},
publisher = {Scuola Normale Superiore, Pisa},
volume = {Ser. 5, 9},
number = {3},
year = {2010},
pages = {635-644},
zbl = {1207.26004},
mrnumber = {2722658},
language = {en},
url = {http://www.numdam.org/item/ASNSP_2010_5_9_3_635_0}
}

Bony, Jean-Michel; Colombini, Ferruccio; Pernazza, Ludovico. On square roots of class $C^m$ of nonnegative functions of one variable. Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Série 5, Volume 9 (2010) no. 3, pp. 635-644. http://www.numdam.org/item/ASNSP_2010_5_9_3_635_0/

[1] D. Alekseevsky, A. Kriegl, P. W. Michor and M. Losik, Choosing roots of polynomials smoothly, Israel J. Math. 105 (1998), 203–233. | MR 1639759 | Zbl 0912.26006

[2] J.-M. Bony, Sommes de carrés de fonctions dérivables, Bull. Soc. Math. France 133 (2005), 619–639. | Numdam | MR 2233698

[3] J.-M. Bony, F. Broglia, F. Colombini and L. Pernazza, Nonnegative functions as squares or sums of squares, J. Funct. Anal. 232 (2006), 137–147. | MR 2200169 | Zbl 1093.26007

[4] J.-M. Bony, F. Colombini and L. Pernazza, On the differentiability class of the admissible square roots of regular nonnegative functions, In: “Phase Space Analysis of Partial Differential Equations”, 45–53, Progr. Nonlinear Differential Equations Appl., Vol. 69, Birkhäuser Boston, Boston, MA, 2006. | MR 2263205 | Zbl 1228.26004

[5] F. Faà Di Bruno, Note sur une nouvelle formule du calcul différentiel, Quarterly J. Pure Appl. Math. 1 (1857), 359–360.

[6] G. Glaeser, Racine carrée d’une fonction différentiable, Ann. Inst. Fourier (Grenoble) 13 (1963), 203–210. | Numdam | MR 163995 | Zbl 0128.27903

[7] A. Kriegl, M. Losik and P.W. Michor, Choosing roots of polynomials smoothly, II, Israel J. Math. 139 (2004), 183–188. | MR 2041790 | Zbl 1071.26009

[8] T. Mandai, Smoothness of roots of hyperbolic polynomials with respect to one-dimensional parameter, Bull. Fac. Gen. Ed. Gifu Univ. 21 (1985), 115–118. | MR 840968