Geodesic
An estimate of the dimension of the null spaces of linear superpositions
Sbornik. Mathematics, Tome 11 (1970) no. 1, pp. 101-114 Cet article a éte moissonné depuis la source Math-Net.Ru

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In this article it is proved that for continuously differentiable functions $f_1(x,y),f_2(x,y),\dots,f_n(x,y)$ a region $U$ of the $x$, $y$ plane can be found such that the dimension of the space of vectors $(\varphi_1(t),\dots,\varphi_n(t))$ for which $\sum_{i=1}^n\varphi_i(f_i(x,y))=0$ in $U$, where $\varphi_i(t)\in L_2$, either equals infinity or else does not exceed the number $(n-1)n/2$. Superpositions of the form $\sum_{i=1}^n\psi_i(f_i(x,y))$ are also shown to be closed and nowhere dense in $L_2$. Bibliography: 3 titles. 
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     title = {An estimate of the dimension of the null spaces of linear superpositions},
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B. L. Fridman. An estimate of the dimension of the null spaces of linear superpositions. Sbornik. Mathematics, Tome 11 (1970) no. 1, pp. 101-114. http://geodesic.mathdoc.fr/item/SM_1970_11_1_a7/

[1] A. G. Vitushkin, G. M. Khenkin, “Lineinye superpozitsii funktsii”, Uspekhi matem. nauk, XXII:1(133) (1967), 77–124

[2] B. L. Fridman, “Uluchshenie gladkosti funktsii v teoreme A. N. Kolmogorova o superpozitsiyakh”, DAN SSSR, 177:5 (1967), 1019–1022 | MR | Zbl

[3] G. E. Shilov, Matematicheskii analiz (vtoroi spets. kurs), Nauka, Moskva, 1965