A Class of Functional Equations (Almost) Characterizing Polynomials on Integral Domains - Ebanks, Bruce R.; de Place Friis, Peter (Sarajevo Journal of Mathematics, 2024)
Let $R$ be an infinite integral domain not of characteristic 2. For a given $ n\geq 2$, suppose functions $f:R\rightarrow R$ and $h:R \rightarrow R$ satisfy \begin{equation*} [x_1,x_2,\ldots,x_n;f]=h(x_1+\cdots+x_n)\prod_{j>i}(x_j-x_i), \end{equation*} where the left side denotes the determinant of the $n\times n$ matrix with row $i$ given by $(1,x_i,x_i^2,\ldots,x_i^{n-2},f(x_i))$. 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 $n\geq 4$. 2000 Mathematics Subject Classification. Primary 39B52