In 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 .

This research aims to propose a new family of one-parameter multi-step iterative methods that combine the homotopy perturbation method with a quadrature formula for solving nonlinear equations. The proposed methods are based on a higher-order convergence scheme that allows for faster and more efficient convergence compared to existing methods. It aims also to demonstrate that the efficiency index of the proposed iterative methods can reach up to $\sqrt[3]{4}\approx 1.587$ and $\sqrt[4]{8}\approx 1.681$, respectively, indicating a high degree of accuracy and efficiency in solving nonlinear equations. To evaluate the effectiveness of the suggested methods, several numerical examples including their performance are provided and compared with existing methods.

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.

In this paper, we consider a generalized monomial $f:\to$ that satisfies the additional equation $f(x)f(y)=0$ for the pairs $(x,y)\in D$, where $D\subset{\mathbb{R}}^{2}$ has a positive planar Lebesgue measure. We prove that $f(x)=0$ for all $x\in\mathbb{R}$. Using analogous arguments, we establish a related statement about the signs of such functions: if a generalized monomial $f$ of an even degree is non-negative on a measurable subset of reals with positive Lebesgue measure, then $f(x)\geq 0$ for every real number $x$. Finally, we extend our results to almost monomial functions.

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.