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

• Published in: Sarajevo journal of mathematics Vol. 1; no. 2; pp. 185 - 196
• Main Authors: Ebanks, Bruce R., De Place Friis, Peter
• Format: Journal Article
• Language: English
• Published: 12.06.2024
• ISSN: 1840-0655; 2233-1964
• DOI: 10.5644/SJM.01.2.05

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