An alternative equation for generalized polynomials involving measure and category constraints

In this paper we consider a generalized polynomial \( f \colon \mathbb{R}^N \to \mathbb{R} \) that satisfies the additional equation \( f(x) f(y) = 0 \) for the pairs \( (x,y) \in D \), where \( D \subseteq \mathbb{R}^{2N} \) has a positive Lebesgue measure or it is a second category Baire set. We prove that \( f(x) = 0 \) for all \( x \in \mathbb{R}^N \). In fact, the first statement is established in a considerably more general setting. Then we formulate statements concerning the signs of generalized monomials \( g \colon \mathbb{R} \to \mathbb{R} \) of even degree that satisfy the inequality \( g(x) g(y) \geq 0 \) for the pairs \( (x,y) \in E \), where \( E \subseteq \mathbb{R}^{2} \) has a positive planar Lebesgue measure or it is a second category Baire set.