TY - JOUR A1 - Kaiser, Tobias T1 - Periods, power series, and integrated algebraic numbers JF - Mathematische Annalen N2 - 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 exponential periods and 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. KW - algebraic geometry KW - algebraic topology KW - algebra KW - associative rings and algebras KW - commutative rings and algebras KW - number theory Y1 - 2024 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:101:1-2024040910285674698609 SN - 0025-5831 SN - 1432-1807 VL - 390 IS - 2 SP - 2043 EP - 2074 PB - Springer Berlin Heidelberg CY - Berlin/Heidelberg ER -