Quadratic polynomials represented by norm forms

TL;DR

This paper proves the Hasse principle and weak approximation for certain quadratic polynomial equations represented by norm forms from quartic extensions, using analytic methods to establish these properties.

Contribution

It establishes the Hasse principle and weak approximation for quadratic polynomials represented by norm forms from quartic extensions, a novel application of analytic techniques.

Findings

Hasse principle holds for the studied equations

Weak approximation is valid in this context

Analytic methods successfully prove these properties

Abstract

The Hasse principle and weak approximation is established for equations of the shape P(t)=N(x_1,x_2,x_3,x_4), where P is an irreducible quadratic polynomial in one variable and N is a norm form associated to a quartic extension of the rationals containing the roots of P. The proof uses analytic methods.