Period (algebraic geometry)

For a more frequently used sense of the word "period" in mathematics, see Periodic function.
The algebraic periods as a subset of the complex numbers.

In mathematics, specifically algebraic geometry, a period or algebraic period[1] is a complex number that can be expressed as an integral of an algebraic function over an algebraic domain. The periods are a class of numbers which includes, alongside the algebraic numbers, many well known mathematical constants such as the number π. Sums and products of periods remain periods, such that the periods form a ring.

Maxim Kontsevich and Don Zagier gave a survey of periods and introduced some conjectures about them.

Periods play an important role in the theory of differential equations and transcendental numbers as well as in open problems of modern arithmetical algebraic geometry.[2] They also appear when computing the integrals that arise from Feynman diagrams, and there has been intensive work trying to understand the connections.[3]

  1. ^ Weisstein, Eric W. "Algebraic Period". mathworld.wolfram.com. Retrieved 2024-09-21.
  2. ^ Kontsevich, Maxim; Zagier, Don (2001). "Periods" (PDF). In Engquist, Björn; Schmid, Wilfried (eds.). Mathematics unlimited—2001 and beyond. Berlin, New York City: Springer. pp. 771–808. ISBN 9783540669135. MR 1852188.
  3. ^ Marcolli, Matilde (2009-07-02). "Feynman integrals and motives". arXiv:0907.0321 [math-ph].