On the Diophantine equations $z^2=f(x)^2 \pm g(y)^2$ concerning Laurent polynomials

By the theory of elliptic curves, we study the nontrivial rational parametric solutions and rational solutions of the Diophantine equations $z^2=f(x)^2 \pm g(y)^2$ for some simple Laurent polynomials $f$ and $g$.
Comment: 7 pages. Submitted for publication. arXiv admin note: substantial text overlap with arXiv:1608.03810