A Class of Functional Equations (Almost) Characterizing Polynomials on Integral Domains

Let R be an infinite integral domain not of characteristic 2. For a given n≥ 2, suppose functions f:R→ R and h:R → R satisfy{equation*}[x_1,x_2,…,x_n;f]=h(x_1+⋯+x_n)∏j>i(x_j-x_i),{equation*}where the left side denotes the determinant of the n× n matrix with row i given by (1,xᵢ,xᵢ²,…,xᵢⁿ⁻²,f(xᵢ)). It is proved that $ Df$ is a polynomial of degree at most n over R, for some D in R. For $n=2$ and $n=3$ the conclusion can be strengthened to take $D=1$, but surprisingly this is not possible for n≥ 4. 2000 Mathematics Subject Classification. Primary 39B52