Norm forms for arbitrary number fields as products of linear polynomials

TL;DR

This paper proves that certain norm form equations over number fields satisfy the Hasse principle and weak approximation when the Brauer-Manin obstruction vanishes, using advanced combinatorial and descent methods.

Contribution

It introduces a novel approach combining additive combinatorics and descent theory to analyze norm form equations over arbitrary number fields.

Findings

Hasse principle holds for the studied norm form equations

Weak approximation is valid under the given conditions

Method integrates Green-Tao techniques with descent theory

Abstract

Let K/Q be a field extension of finite degree and let P(t) be a polynomial over Q that splits into linear factors over Q. We show that any smooth model of the affine variety defined by the equation N_{K/Q} (k) = P(t) satisfies the Hasse principle and weak approximation whenever the Brauer-Manin obstruction is empty. Our proof is based on a combination of methods from additive combinatorics due to Green-Tao and Green-Tao-Ziegler, together with an application of the descent theory of Colliot-Th\'el\`ene and Sansuc.