Saved in:
| Main Author: | |
|---|---|
| Format: | Preprint |
| Published: |
2022
|
| Subjects: | |
| Online Access: | https://arxiv.org/abs/2211.01269 |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866910311655669760 |
|---|---|
| author | Kaiser, Tobias |
| author_facet | Kaiser, Tobias |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2211_01269 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Periods, Power Series, and Integrated Algebraic Numbers Kaiser, Tobias Logic Algebraic Geometry Number Theory 03C64, 11J81, 13J05, 13J30, 14P10, 32B20, 51M25 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. |
| title | Periods, Power Series, and Integrated Algebraic Numbers |
| topic | Logic Algebraic Geometry Number Theory 03C64, 11J81, 13J05, 13J30, 14P10, 32B20, 51M25 |
| url | https://arxiv.org/abs/2211.01269 |