An alternative equation for generalized monomials

Abstract

In this paper we consider a generalized monomial or polynomial f : mathbb {R}rightarrow mathbb {R} that satisfies the additional equation f(x) f(y) = 0 for the pairs (x,y) in D ,, where D subseteq {mathbb {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 D={(p(t),q(t))|t∈R}\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} D = \\{\\, (p(t),q(t)) \\,\\vert \\, t \\in \\mathbb {R}\\,\\} \\end{aligned}$$\\end{document}or f is a generalized monomial and there exists a positive rational m fulfilling D={(x,y)∈R2|x2-my2=1},\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} D = \\{\\, (x,y) \\in {\\mathbb {R}}^{2} \\,\\vert \\, x^2 - m y^2 = 1 \\,\\}, \\end{aligned}$$\\end{document}we prove that f(x) = 0 for all x in mathbb {R},.