Linear functional equations with algebraic parameters (Q2875459)

The author investigates the linear functional equation NEWLINE\[NEWLINE\sum_{i=1}^{n}a_{i}f(b_{i}x+c_{i}y)=0NEWLINE\]NEWLINE where \(a_{i}\in\mathbb C\) (the set of all complex numbers), \(b_{i}\in K\) (a subfield of \(\mathbb{C}\)) are given \((i=1,\dots, n)\), and \(f:\mathbb C\to\mathbb C\) is the unknown function under the assumptions on the parameters when the solutions are generalized polynomials. As a typical result, he proves the following statement: if the parameters \(b_{i},c_{i}\) \((i=1, \dots, n)\) are algebraic numbers (over the rationals) satisfying some natural assumptions, \(K\) is the smallest subfield of \(\mathbb C\) containing the parameters \(b_{1}, \dots, b_{n}, c_{1}, \dots, c_{n}\) and \(f:K\to\mathbb C\) is a solution of degree at most \(k\) then \(f\) is the linear combination of products \(k\) injective homomorphisms of \(K\) which products are also solutions of the above equation.