{"paper":{"title":"Rational solutions of certain Diophantine equations involving norms","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Maciej Ulas","submitted_at":"2013-05-27T14:44:30Z","abstract_excerpt":"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{"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1305.6242","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}