Almost universality of a sum of norms

TL;DR

This paper proves that a specific class of polynomials involving perfect powers and norm forms from abelian number fields can represent all but finitely many natural numbers, using advanced number theory techniques.

Contribution

It introduces a new class of polynomials with a sum of norms that are almost universal, extending previous results in number theory.

Findings

The polynomial represents all but finitely many natural numbers.

The circle method and local class field theory are effectively combined.

The approach generalizes previous universality results for norm forms.

Abstract

In this paper the author considers a particular type of polynomials with integer coefficients, consisting of a perfect power and two norm forms of abelian number fields with coprime discriminants. It is shown that such a polynomial represents every natural number with only finitely many exceptions. The circle method is used, and the local class field theory plays a central role in estimating the singular series.