A Class of Functional Equations (Almost) Characterizing Polynomials on Integral Domains
DOI:
https://doi.org/10.5644/SJM.01.2.05Keywords:
Characterization of polynomials, divided difference, functional equations, integral domainsAbstract
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
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