In algebraic geometry, a period is a number that can be expressed as an integral of an algebraic function over an algebraic domain. Sums and products of periods remain periods, so the periods form a ring. gave a survey of periods and introduced some conjectures about them.
Definition
A real number is called a period if it is the difference of volumes of regions of Euclidean space given by polynomialinequalities with rational coefficients. More generally a complex number is called a period if its real and imaginary parts are periods. Periods are numbers that arise as integrals of algebraic functions over domains that are described by algebraic equations or by inequalities with rational coefficients. Periods can be defined as complex numbers whose real and imaginary parts are values of absolutely convergentintegrals of rational functions with rational coefficients, over domains in given by polynomial inequalities with rational coefficients. The coefficients of the rational functions and polynomials can be generalised to algebraic numbers since integrals and irrational algebraic numbers are expressible in terms of areas of suitable domains.
Examples
Besides the algebraic numbers, the following numbers are known to be periods:
An example of a real number that is not a period is given by Chaitin's constant Ω. Any other non-computable number also gives an example of a real number that is not a period. Currently there are no natural examples of computable numbers that have been proved not to be periods, however it is possible to construct artificial examples. Plausible candidates for numbers that are not periods include e, 1/, and Euler–Mascheroni constant γ.
Properties and motivation
The periods are intended to bridge the gap between the algebraic numbers and the transcendental numbers. The class of algebraic numbers is too narrow to include many common mathematical constants, while the set of transcendental numbers is not countable, and its members are not generally computable. The set of all periods is countable, and all periods are computable, and in particular definable.
Conjectures
Many of the constants known to be periods are also given by integrals of transcendental functions. Kontsevich and Zagier note that there "seems to be no universal rule explaining why certain infinite sums or integrals of transcendental functions are periods". Kontsevich and Zagier conjectured that, if a period is given by two different integrals, then each integral can be transformed into the other using only the linearity of integrals, changes of variables, and the Newton-Leibniz formula . A useful property of algebraic numbers is that equality between two algebraic expressions can be determined algorithmically. The conjecture of Kontsevich and Zagier would imply that equality of periods is also decidable: inequality of computable reals is known recursively enumerable; and conversely if two integrals agree, then an algorithm could confirm so by trying all possible ways to transform one of them into the other one. It is not expected that Euler's numbere and Euler-Mascheroni constant γ are periods. The periods can be extended to exponential periods by permitting the product of an algebraic function and the exponential function of an algebraic function as an integrand. This extension includes all algebraic powers of e, the gamma function of rational arguments, and values of Bessel functions. If, further, Euler's constant γ is added as a new period, then according to Kontsevich and Zagier "all classical constants are periods in the appropriate sense".
