Primes represented by incomplete norm forms

TL;DR

This paper proves that certain incomplete norm forms in algebraic number fields take infinitely many prime values under specific degree conditions, extending previous prime value results to new polynomial forms.

Contribution

It establishes asymptotic prime counts for incomplete norm forms and demonstrates infinite prime values in special cases, using advanced sieve methods and geometric estimates.

Findings

Incomplete norm forms take the expected number of prime values when degree conditions are met.

In the case of radical extensions, these forms produce infinitely many primes if degree is sufficiently large.

The proof combines Harman's sieve with geometry of numbers and algebraic geometry techniques.

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.