Oscillatory integrals with phases arising from algebraic number fields

TL;DR

This paper develops a theory for oscillatory integrals with phases derived from algebraic number fields and applies it to a singular integral related to Tarry's problem for algebraic integers.

Contribution

It introduces a new framework for analyzing oscillatory integrals with phases from algebraic number fields and applies it to a classical problem in algebraic integers.

Findings

Established a theory for oscillatory integrals with algebraic number field phases

Applied the theory to Tarry's problem for algebraic integers

Provided insights into singular integrals in algebraic number theory

Abstract

We develop a theory of oscillatory integrals whose phase is given by the trace of a polynomial over an algebraic number field. We present an application to the singular integral for a version of Tarry's problem for algebraic integers.