Rational solutions of certain Diophantine equations involving norms
Tom 165 / 2014
Streszczenie
We present some results concerning the unirationality of the algebraic variety $\mathcal {S}_{f}$ given by the equation $$ N_{K/k}(X_{1}+\alpha X_{2}+\alpha ^2 X_{3})=f(t), $$ 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 $\deg f=4$ and $\mathcal {S}_{f}$ contains a $k$-rational point $(x_{0},y_{0},z_{0},t_{0})$ with ${f(t_{0})\not =0}$, then $\mathcal {S}_{f}$ 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 $\mathcal {S}_{f}$ (with a non-trivial $k$-rational point) is proved for any polynomial $f$ of degree 6 with $f$ not equivalent to a polynomial $h$ satisfying $h(t)=h(\zeta _{3}t)$, 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 a root of the 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 $a_{1}a_{4}\not =0$.
