On the characteristic polynomials of linear functional equations (Q519939)

Consider the linear functional equation \[ \sum\limits_{i=0}^n a_if(b_ix+(1-b_i)y)=0\qquad(x,y \in I) \] where \(I\) is a nonempty open real interval, \(0<b_0<b_1<\dots<b_n<1\) and \(a_0,a_1,\dots,a_n\) are given nonzero real numbers. According to the results of \textit{L. Székelyhidi} [Publ. Math. 29, 19--28 (1982; Zbl 0508.39014); Convolution type functional equations on topological abelian groups, Singapore etc.: World Scientific (1991; Zbl 0748.39003)] any solution \(f:I\to\mathbb{R}\) of the above equation must be a generalized polynomial of the form \[ f(x)=\sum\limits_{k=1}^{n-1}A_k(x,\dots,x)+A_0\qquad(x\in I) \] where \(A_k:\mathbb{R}^k\to\mathbb{R}\) is a symmetric \(k\)-additive function for any \(k=1,\dots,n-1\), \(A_0\in \mathbb R.\) From the author's abstract: ``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 polynomial of a linear functional equation such that the algebraic properties of the roots allows us to conclude the existence of non-trivial solutions.''