Rational solutions of certain Diophantine equations involving norms

In this note we present some results concerning the unirationality of the algebraic variety ${\mathcal{S}}_{\mathcal{f}}$ 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 $\text{op}degf=4$ and the variety ${\mathcal{S}}_{\mathcal{f}}$ contains a $k$-rational point $(x_0,y_0,z_0,t_0)$ with $f(t_0)\mathrm{\ne }0$, then ${\mathcal{S}}_{\mathcal{f}}$ 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 ${\mathcal{S}}_{\mathcal{f}}$ (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 $h(t)\mathrm{\ne }h({\zeta }_{3}t)$, 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 $h(x)=x^3+ax+b\in k[x]$,providedthat$f(t)=t^6+a_{4}t^4+a_{1}t+a_{0}\in k[t]$with$a_{1}a_{4}\neq 0$.