Polynomials represented by norm forms via the beta sieve

TL;DR

This paper proves the Hasse principle for a broad class of polynomials represented by norm forms over number fields, extending previous results to higher degrees using the beta sieve method.

Contribution

It generalizes Irving's argument with the beta sieve to establish the Hasse principle for polynomials with multiple linear, quadratic, or cubic factors over various number fields.

Findings

Proved the Hasse principle for a wide family of polynomials and number fields.

Established new cases of a conjecture on locally split polynomial values.

Applied sieve techniques to problems in rational points and fibrations.

Abstract

A central question in Arithmetic geometry is to determine for which polynomials $f \in \mathbb{Z}[t]$ and which number fields $K$ the Hasse principle holds for the affine equation $f(t) = N_{K/\mathbb{Q}}(\boldsymbol{x}) \neq 0$. Whilst extensively studied in the literature, current results are largely limited to polynomials and number fields of low degree. In this paper, we establish the Hasse principle for a wide family of polynomials and number fields, including polynomials that are products of arbitrarily many linear, quadratic or cubic factors. The proof generalises an argument of Irving, which makes use of the beta sieve of Rosser and Iwaniec. As a further application of our sieve results, we prove new cases of a conjecture of Harpaz and Wittenberg on locally split values of polynomials over number fields, and discuss consequences for rational points in fibrations.