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≥ 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,x_i,x_i^2,…,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≥ 4. 2000 Mathematics Subject Classification. Primary 39B52