On the Diophantine equations z2=f(x)2±f(y)2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$z^2=f(x)^2 \pm f(y)^2$$\end{document} involving quartic polynomials

By the theory of elliptic curves, we prove that the Diophantine equations z2=f(x)2±f(y)2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$z^2=f(x)^2 \pm f(y)^2$$\end{document} have infinitely many rational solutions for some quartic polynomials, which gives a positive answer to Question 4.3 of Ulas and Togbé (Publ Math Debrecen 76(1–2):183–201, 2010) for quartic polynomials.