Periods, Power Series, and Integrated Algebraic Numbers

TL;DR

This paper explores the algebraic structure of periods, showing they belong to a countable real closed field generated by algebraic power series and their integrals, and introduces exponential integrated algebraic numbers including the Euler constant.

Contribution

It constructs a new system of rings of power series that contain all periods and defines exponential integrated algebraic numbers, linking periods with o-minimality and constants like Euler's.

Findings

All periods are contained in a countable real closed field.

The Euler constant is an exponential integrated algebraic number.

The constructed rings extend the period ring with algebraic and exponential operations.

Abstract

Periods are defined as integrals of semialgebraic functions defined over the rationals. Periods form a countable ring not much is known about. Examples are given by taking the antiderivative of a power series which is algebraic over the polynomial ring over the rationals and evaluate it at a rational number. We follow this path and close these algebraic power series under taking iterated antiderivatives and nearby algebraic and geometric operations. We obtain a system of rings of power series whose coefficients form a countable real closed field. Using techniques from o-minimality we are able to show that every period belongs to this field. In the setting of o-minimality we define exponential integrated algebraic numbers and show that the Euler constant is an exponential integrated algebraic number. Hence they are a good candiate for a natural number system extending the period ring and containing important mathematical constants.