Characterization of polynomials and divided difference (Q1917763)

The authors offer two additional proofs of a result by \textit{J. Schwaiger} [Aequationes Math. 48, No. 2-3, 317-323 (1994; Zbl 0810.39007)] that the \((n-1)\)-st divided difference (with variable spans) of \(f\) is a function of the sum of its \(n\) variables iff \(f\) is a polynomial of degree at most \(n\); and \(g\) is linear. The variables and function values are here in the set of all reals (in Schwaiger's paper they are in any field with sufficiently many elements and with characteristic different from 2).