An alternative equation for generalized monomials

Abstract

AbstractIn this paper we consider a generalized monomial or polynomial $$ f : \mathbb {R}\rightarrow \mathbb {R}$$ f : R → R that satisfies the additional equation $$ f(x) f(y) = 0 $$ f ( x ) f ( y ) = 0 for the pairs $$ (x,y) \in D \,$$ ( x , y ) ∈ D , where $$ D \subseteq {\mathbb {R}}^{2} $$ 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 $$\begin{aligned} D = \{\, (p(t),q(t)) \,\vert \, t \in \mathbb {R}\,\} \end{aligned}$$ D = { ( p ( t ) , q ( t ) ) | t ∈ R } or f is a generalized monomial and there exists a positive rational m fulfilling $$\begin{aligned} D = \{\, (x,y) \in {\mathbb {R}}^{2} \,\vert \, x^2 - m y^2 = 1 \,\}, \end{aligned}$$ D = { ( x , y ) ∈ R 2 | x 2 - m y 2 = 1 } , we prove that $$ f(x) = 0 $$ f ( x ) = 0 for all $$ x \in \mathbb {R}\,$$ x ∈ R .