Using the theory of elliptic curves, we study the nontrivial rational (parametric) solutions of the Diophantine equations $z^2=f(x)^2 \pm f(y)^2$ for some simple Laurent polynomials $f(x)$.
"On the Diophantine equations $z^2=f(x)^2 \pm f(y)^2$ involving Laurent polynomials." Funct. Approx. Comment. Math. 62 (2) 187 - 201, June 2020. https://doi.org/10.7169/facm/1766