A divided-difference characterization of polynomials over a general field
Generalizing the characterization of real quadratic functions \(f\) by the property \((f(x_{1})-f(x_{2}))/\) \((x_{1}-x_{2})=h(x_{1}+x_{2})\), this reviewer has proved [Math. Mag. 58, 42-45 (1985; Zbl 0571.39005)] that this equation characterizes the polynomials \(f\) of second degree also among functions mapping an arbitrary field of characteristic different from 2 into itself. The generalization that polynomials of \(n\)th degree are characterized by the property that the \((n-1)\)st divided difference \(f[x_{1},\dots,x_{n}]\) depends only upon \(x_{1}+\dots+x_{n}\) has been proved by \textit{J. Schwaiger} [Aequationes Math. 48, No. 2-3, 317-323 (1994; Zbl 0810.39007)] in the case where the number of elements of the field is at least \(8n-14\). The present authors eliminate this restriction. Furthermore, they consider the case of characteristic 2, pointing out also an error in a counter-example in the reviewer's paper quoted above.
scientific article; zbMATH DE number 1130461