Cubic polynomials represented by norm forms

A. J. Irving

Published 2013-10-23, updated 2015-04-01Version 3

We show that for an irreducible cubic $f\in\mathbb Z[x]$ and a full norm form $\mathbf N(x_1,\ldots,x_k)$ for a number field $K/\mathbb Q$ satisfying certain hypotheses the variety $f(t)=\mathbf N(x_1,\ldots,x_k)\ne 0$ satisfies the Hasse principle. Our proof uses sieve methods.