Rational points on certain elliptic surfaces

Let ${\mathcal{E}}_f:y^2=x^3+f(t)x$, where $f\in \text{Q}[t]\setminus \text{Q}$, and let us assume that $\text{op}degf\le 4$. In this paper we prove that if $\op{deg}f\leq 3$,thenthereexistsarationalbasechange$t\mapsto\phi(t)$ such that on the surface ${\mathcal{E}}_{\mathcal{f}\circ \mathcal{\varphi }}$ there is a non-torsion section. A similar theorem is valid in case when $\text{op}degf=4$ and there exists $t_0\in \text{Q}$ such that infinitely many rational points lie on the curve $E_{t_0}:y^2=x^3+f(t_0)x$. In particular, we prove that if $\text{op}degf=4$ and $f$ is not an even polynomial, then there is a rational point on ${\mathcal{E}}_{\mathcal{f}}$. Next, we consider a surface ${\mathcal{E}}^{\mathcal{g}}:{\mathcal{y}}^{\mathcal{2}}={\mathcal{x}}^{\mathcal{3}}+\mathcal{g}(\mathcal{t})$, where $g\in \text{Q}[t]$ is a monic polynomial of degree six. We prove that if the polynomial $g$ is not even, there is a rational base change $t\mapsto \psi (t)$ such that on the surface ${\mathcal{E}}^{\mathcal{g}\circ \mathcal{\psi }}$ there is a non-torsion section. Furthermore, if there exists $t_0\in \text{Q}$ such that on the curve $E^{t_0}:y^2=x^3+g(t_0)$ there are infinitely many rational points, then the set of these $t_0$ is infinite. We also present some results concerning diophantine equation of the form $x^2-y^3-g(z)=t$, where $t$ is a variable.