On the characteristic polynomials of linear functional equations

The solutions of a linear functional equation are typically generalized polynomials. The existence of their non-trivial monomial terms strongly depends on the algebraic properties of some related families of parameters. In extremal cases (the parameters are algebraic numbers or the parameters form an algebraically independent system) we have elegant methods to decide the existence of non-trivial solutions. In this paper we are going to extend and unify the treatment of the existence problem by introducing the characteristic polynomials of a linear functional equation such that the algebraic properties of the roots allows us to conclude the existence of non-trivial solutions.