An Alternative Equation for Generalized Polynomials of Degree Two

Abstract 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 ⊆ ℝ 2 is given by some algebraic condition. In the particular cases when there exists a positive rational m fulfilling D = { ( x , y ) ∈ ℝ 2 | x 2 - m y 2 = 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.