Primes represented by incomplete norm forms

Abstract

Let $K=\mathbb{Q}(\omega)$ with $\omega$ the root of a degree $n$ monic irreducible polynomial $f\in\mathbb{Z}[X]$. We show the degree $n$ polynomial $N(\sum_{i=1}^{n-k}x_i\omega^{i-1})$ in $n-k$ variables formed by setting the final $k$ coefficients to 0 takes the expected asymptotic number of prime values if $n\ge 4k$. In the special case $K=\mathbb{Q}(\sqrt[n]{\theta})$, we show $N(\sum_{i=1}^{n-k}x_i\sqrt[n]{\theta^{i-1}})$ takes infinitely many prime values provided $n\ge 22k/7$. 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.