Rational solutions of certain Diophantine equations involving norms

In this note we present some results concerning the unirationality of the algebraic variety $\cal{S}_{f}$ given by the equation \begin{equation*} N_{K/k}(X_{1}+\alpha X_{2}+\alpha^2 X_{3})=f(t), \end{equation*} where $k$ is a number field, $K=k(\alpha)$, $\alpha$ is a root of an irreducible polynomial $h(x)=x^3+ax+b\in k[x]$ and $f\in k[t]$. We are mainly interested in the case of pure cubic extensions, i.e. $a=0$ and $b\in k\setminus k^{3}$. We prove that if $\op{deg}f=4$ and the variety $\cal{S}_{f}$ contains a $k$-rational point $(x_{0},y_{0},z_{0},t_{0})$ with $f(t_{0})\neq 0$, then $\cal{S}_{f}$ is $k$-unirational. A similar result is proved for a broad family of quintic polynomials $f$ satisfying some mild conditions (for example this family contains all irreducible polynomials). Moreover, the unirationality of $\cal{S}_{f}$ (with non-trivial $k$-rational point) is proved for any polynomial $f$ of degree 6 with $f$ not equivalent to the polynomial $h$ satisfying the condition $h(t)\neq h(\zeta_{3}t)$, where $\zeta_{3}$ is the primitive third root of unity. We are able to prove the same result for an extension of degree 3 generated by the root of polynomial $h(x)=x^3+ax+b\in k[x]$, provided that $f(t)=t^6+a_{4}t^4+a_{1}t+a_{0}\in k[t]$ with $a_{1}a_{4}\neq 0$.