An Alternative Equation for Generalized Polynomials of Degree Two

In this paper we consider a generalized polynomial f: ℝ → ℝ of degree two that satisfies the additional equation f(x)f(y) = 0 for the pairs (x, y) ∈ D, where D ⊆ ℝ² is given by some algebraic condition. In the particular cases when there exists a positive rational m fulfilling D={ (x,y)∈ℝ2|x2-my2=1 }, D = \left\{ {\left( {x,y} \right) \in \mathbb{R}{^2}|{x^2} - m{y^2} = 1} \right\}, we prove that f(x) = 0 for all x ∈ ℝ.