Published online by Cambridge University Press: 06 February 2020
Let $K=\mathbb{Q}(\unicode[STIX]{x1D714})$ with $\unicode[STIX]{x1D714}$ the root of a degree $n$ monic irreducible polynomial $f\in \mathbb{Z}[X]$. We show that the degree $n$ polynomial $N(\sum _{i=1}^{n-k}x_{i}\unicode[STIX]{x1D714}^{i-1})$ in $n-k$ variables takes the expected asymptotic number of prime values if $n\geqslant 4k$. In the special case $K=\mathbb{Q}(\sqrt[n]{\unicode[STIX]{x1D703}})$, we show that $N(\sum _{i=1}^{n-k}x_{i}\sqrt[n]{\unicode[STIX]{x1D703}^{i-1}})$ takes infinitely many prime values, provided $n\geqslant 22k/7$.
Our proof relies on using suitable ‘Type I’ and ‘Type II’ estimates in Harman’s sieve, which are established in a similar overall manner to the previous work of Friedlander and Iwaniec on prime values of $X^{2}+Y^{4}$ and of Heath-Brown on $X^{3}+2Y^{3}$. Our proof ultimately relies on employing explicit elementary estimates from the geometry of numbers and algebraic geometry to control the number of highly skewed lattices appearing in our final estimates.