Characterization of polynomials as solutions of certain functional equations

In this paper we present several characterizations of ordinary polynomials as the solution sets of certain functional equations related to the equation∑i=0mfi(bix+ciy)=∑i=1nai(y)vi(x), where x,y∈Rd and bi,ci∈GLd(C), whose solution set is, typically, formed by exponential polynomials. Some of these equations are important because of their connection with the Characterization Problem of distributions in Probability Theory.