Alienation of two general linear functional equations

Abstract

We study the alienation problem for two general linear equations i.e. we compare the solutions of the system of equations ∑i=1nαif(pix+qiy)=0∑j=1mβjg(sjx+tjy)=0\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$\\begin{aligned} \\left\\{ \\begin{array}{l}\\sum _{i=1}^n\\alpha _if(p_ix+q_iy)=0\\\\ \\sum _{j=1}^m\\beta _jg(s_jx+t_jy)=0\\end{array}\\right. \\end{aligned}$$\\end{document}with the solutions of the single equation ∑i=1nαif(pix+qiy)=∑j=1mβjg(sjx+tjy).\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$\\begin{aligned} \\sum _{i=1}^n\\alpha _if(p_ix+q_iy)=\\sum _{j=1}^m\\beta _jg(s_jx+t_jy). \\end{aligned}$$\\end{document}To this end we introduce the notion of l-alienation—alienation in the class of monomial functions of order l. We use our results among others to study the alienation properties of two monomial functional equations.