Geodesic
On the $id$-decompositions of the class $P_k$ over precomplete classes
Diskretnaya Matematika, Tome 5 (1993) no. 2, pp. 98-110 
S. S. Marchenkov. On the $id$-decompositions of the class $P_k$ over precomplete classes. Diskretnaya Matematika, Tome 5 (1993) no. 2, pp. 98-110. http://geodesic.mathdoc.fr/item/DM_1993_5_2_a7/
@article{DM_1993_5_2_a7,
     author = {S. S. Marchenkov},
     title = {On the $id$-decompositions of the class~$P_k$ over precomplete classes},
     journal = {Diskretnaya Matematika},
     pages = {98--110},
     year = {1993},
     volume = {5},
     number = {2},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/DM_1993_5_2_a7/}
}
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We consider a representation of functions $f(x_1,\cdots,x_n)$ in $P_k$ in the form $$ g(x_1,\dots,x_m,F^1_2,\dots,F^1_m,\dots,F^m_1,\dots,F^m_{m-1}), $$ where $2\le m\le n$ and $F^i_j=f(x_1,\dots,x_{j-1},x_i,x_{j+1},\dots,x_n)$ for $i\ne j$. We investigate the possibility of such representations with $g$ belonging to classes that are precomplete in $P_k$. We give upper bounds on the parameter $m$ in the representation.