On extension of polynomial functions (Q1895192)

The author studies the solutions of the functional equation \[ \sum^{n+1}_{j=0} (-1)^{n+1 -j} {n+1\choose j} f\Bigl(\bigl( 1- {j\over n+1}\bigr) x+{j\over n+1} y\Bigr)=0, \] where the unknown function \(f\) is a mapping from a \(Q\)-convex subset \(D\) of a linear space \(X\) over the field of rationals into another linear space \(Y\) over \(Q\). By means of some elegant new tools it is proved that any such \(f\) admits an extension \(F:X\to Y\) of the form \(F(x)=A^0+A^1(x)+\cdots+A^n(x)\), where \(A^0\in Y\) and the maps \(A_k:X^k\to Y\) such that \(A_k(x,\ldots, x)=A^k(x)\) are \(k\)-additive and symmetric. The uniqueness problem of the extension \(F\) is discussed.