An alternative equation for generalized polynomials involving measure and category constraints
Year: 2025
Type: article
Abstract: In this paper we consider a generalized polynomial $$ f \colon \mathbb{R}^N \to \mathbb{R} $$ f : R N → 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 $$ D \subseteq \mathbb{R}^{2N} $$ D ⊆ R 2 N has a positive Lebesgue measure or it is a second category Baire set. We prove that $$ f(x) = 0 $$ f ( x ) = 0 for all $$ x \in \mathbb{R}^N $$ x ∈ 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 $$ g \colon \mathbb{R} \to \mathbb{R} $$ g : R → R of even degree that satisfy the inequality $$ g(x) g(y) \geq 0 $$ g ( x ) g ( y ) ≥ 0 for the pairs $$ (x,y) \in E $$ ( x , y ) ∈ E , where $$ E \subseteq \mathbb{R}^{2} $$ E ⊆ R 2 has a positive planar Lebesgue measure or it is a second category Baire set.
Institution: University of Debrecen
Language: English
Subfield: Mathematical Physics
Field: Mathematics
Domain: Physical Sciences
Open Access status: hybrid
Estimated APC paid: $3,090
OpenAlex ID: W4410547264