Geodesic
Homogeneous partial differential equations for superpositions of indeterminate functions of several variables
Izvestiya. Mathematics , Tome 73 (2009) no. 1, pp. 31-46.

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We determine essentially all partial differential equations satisfied by superpositions of tree type and of a further special type. These equations represent necessary and sufficient conditions for an analytic function to be locally expressible as an analytic superposition of the type indicated. The representability of a real analytic function by a superposition of this type is independent of whether that superposition involves real-analytic functions or $C^{\rho}$-functions, where the constant $\rho$ is determined by the structure of the superposition. We also prove that the function $u$ defined by $u^n=xu^a+yu^b+zu^c+1$ is generally non-representable in any real (resp. complex) domain as $f\bigl(g(x,y),h(y,z)\bigr)$ with twice differentiable $f$ and differentiable $g$, $h$ (resp. analytic $f$, $g$, $h$).
Keywords: essentially all PDEs, rooted trees, Hilbert's 13th problem, minors.
Mots-clés : superposition 
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K. Asai. Homogeneous partial differential equations for superpositions of indeterminate functions of several variables. Izvestiya. Mathematics , Tome 73 (2009) no. 1, pp. 31-46. http://geodesic.mathdoc.fr/item/IM2_2009_73_1_a3/

[1] P. S. Alexandrov (ed.), Die Hilbertschen Probleme, Second edition, Ostwalds Klassiker Exakt. Wiss., 252, Geest Portig, Leipzig, 1979 | MR | MR | Zbl | Zbl

[2] D. Hilbert, Gesammelte Abhandlungen. Bd. III: Analysis, Grundlagen der Mathematik, Physik, Verschiedenes. Nebst einer Lebensgeschichte, Springer-Verlag, Berlin, 1935 | MR | Zbl

[3] V. I. Arnol'd, “On functions of three variables”, Amer. Math. Soc. Transl. Ser. 2, 28 (1963), 51–54 | MR | Zbl

[4] V. I. Arnol'd, “On the representation of continuous functions of three variables by superpositions of continuous functions of two variables”, Amer. Math. Soc. Transl. Ser. 2, 28 (1963), 61–147 | MR | Zbl

[5] A. N. Kolmogorov, “On the representation of continuous functions of several variables by superpositions of continuous functions of a smaller number of variables”, Amer. Math. Soc. Transl. Ser. 2, 17 (1961), 369–373 | MR | Zbl

[6] A. N. Kolmogorov, “On the representation of continuous functions of many variables by superposition of continuous functions of one variable and addition”, Amer. Math. Soc. Transl. Ser. 2, 28 (1963), 55–59 | MR | Zbl | Zbl

[7] A. N. Kolmogorov, Selected works of A. N. Kolmogorov. Vol. I. Mathematics and mechanics, Math. Appl. (Soviet Ser.), 25, Kluwer Acad. Publ., Dordrecht, 1991 | MR | MR | Zbl | Zbl

[8] V. I. Arnol'd, “From Hilbert's superposition problem to dynamical systems”, Amer. Math. Monthly, 111:7 (2004), 608–624 | DOI | MR

[9] K. Asai, “Taylor expansion of implicit functions defined by linear equations of variables”, Kodai Math. J., 24:1 (2001), 31–35 | DOI | MR | Zbl

[10] | Zbl