The octahedron equation implies the cube equation: an elementary proof
It has been proved in the paper by \textit{J. Aczél}, the reviewer, \textit{M. A. McKiernan} and \textit{G. N. Sakovič} [ibid. 1, 37-53 (1968; Zbl 0157.461)] that the following two functional equations are equivalent without any regularity assumption: \[ (1)\quad f(x+t,y+t)+f(x-t,y+t)+f(x- t,y-t)+f(x+t,y-t)=4f(x,y) \] and (2) \(f(x+t,y)+f(x,y+t)+f(x-t,y)+f(x,y- t)=4f(x,y)\), where \(f:R^ 2\to R\) and x,y,t are real variables. It has been conjectured by D. Z. Djoković and the reviewer that the n- dimensional cube functional equation denoted by \((C_ n)\) which is a generalization of(1) and the n-dimensional octahedron functional equation denoted by \((O_ n)\) which is a generalization of (2) are equivalent for all \(n\geq 1\). \textit{L. Sweet} [ibid. 22, 29-38 (1981; Zbl 0475.39016)] proved that \((O_ n)\) implies \((C_ n)\) for all \(n\geq 1\). In the present note the authors give another proof of this theorem by using Newton's formulas on sums of powers of roots for algebraic equations.
n-dimensional octahedron functional equation
scientific article; zbMATH DE number 3983780