Algebraic Period

In algebra, a period is a number that can be written an integral of an algebraic function over an algebraic domain. More specifically, a period is a real number


where P is a polynomial and Q is a rational function on R^n with rational coefficients.

Periods (which are countable) were defined to fill in a gap between algebraic numbers (which do not contain many mathematical constants) and the transcendental numbers (which are not countable). In particular, any algebraic number is a period and any number that is not a period is a transcendental number (Waldschmidt 2006), so there is a "gap" between those two statements in the sense that algebraic periods may be algebraic or transcendental. As a result, Kontsevich and Zagier (2001) propose their Principle 1: "Whenever you meet a new number, and have decided (or convinced yourself) that it is transcendental, try to figure out whether it is a period."

Periods form a ring since sums and products of periods are also periods. However, this class of numbers is larger and less well understood than the ring of algebraic numbers. However, its elements are constructible and it is conjectured that equality of any two numbers which have been expressed as periods can be verified. Most of the important constants of mathematics belong to the class of periods (Kontsevich and Zagier 2001).

Examples of periods include


zeta(n) for n a positive integer where zeta(s) is the Riemann zeta function, Gamma(p/q) for p,q positive integers, and [Gamma(p/q)]^q.

Chaitin's constant Omega is not a period.

It is not known if e, 1/pi, or the Euler-Mascheroni constant gamma are periods, though it it conjectured that e and gamma are not.

See also

Algebtaic Number, Transcendental Number

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Belkale, P. and Brosnan, P. "Periods and Igusa Local Zeta Functions." Int. Math. Res. Not. 49, 2655-2670, 2003.Kontsevich, M. and Zagier, D. "Periods." In athematics UnlimitedÑ-2001 and Beyond (Ed. B. Engquist and W. Schmid). Berlin: Springer, pp. 771-808, 2001.Marcolli, M. "Feynman Integrals and Motives." Europ. Congress Math. Eur. Math. Soc. Zürich, pp. 293-332, 2010.Waldschmidt, M. "Transcendence of Periods: The State of the Art." Pure Appl. Math. Quart. 2, 435-463, 2006.Yoshinaga, M. "Periods and Elementary Real Numbers." 03 May 2008.

Cite this as:

Weisstein, Eric W. "Algebraic Period." From MathWorld--A Wolfram Web Resource.

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