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In this paper we consider generalized monomial functions f , g : F → C (of possibly different degree) that also fulfill f (P (x)) = Q (g (x)) x ∈ F , where P ∈ F [ x ] and Q ∈ C [ x ] are given (classical) polynomials. [ABSTRACT FROM AUTHOR]

In this paper we consider a generalized monomial or polynomial f : R → R that satisfies the additional equation f (x) f (y) = 0 for the pairs (x , y) ∈ D , where D ⊆ R 2 is given by some algebraic condition. In the particular cases when f is a generalized polynomial and there exist non-constant regular polynomials p and q that fulfill D = { (p (t) , q (t)) | t ∈ R } or f is a generalized monomial and there exists a positive rational m fulfilling D = { (x , y) ∈ R 2 | x 2 - m y 2 = 1 } , we prove that f (x) = 0 for all x ∈ R . [ABSTRACT FROM AUTHOR]