In mathematics, a period, or algebraic period, 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, so that the periods form a ring, commonly denoted
P
{\displaystyle {\mathcal {P}}}
.
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 arithmetic geometry. They also appear when computing the integrals that arise from Feynman diagrams, and there has been intensive work trying to understand the connections.
Contents
Definition
A number
α
{\displaystyle \alpha }
is a period if it can be expressed as an integral of the form
α
=
∫
P
(
x
1
,
…
,
x
n
)
≥
0
Q
(
x
1
,
…
,
x
n
)
d
x
1
…
d
x
n
,
{\displaystyle \alpha =\int _{P(x_{1},\ldots ,x_{n})\geq 0}Q(x_{1},\ldots ,x_{n})\ \mathrm {d} x_{1}\ldots \mathrm {d} x_{n},}
where
P
{\displaystyle P}
is a polynomial and
Q
{\displaystyle Q}
a rational function on
R
n
{\displaystyle \mathbb {R} ^{n}}
with rational coefficients. A complex number is a period if its real and imaginary parts are periods.
An equivalent definition allows
P
{\displaystyle P}
and
Q
{\displaystyle Q}
to be algebraic functions. The coefficients of the rational functions and polynomials can also be generalized to algebraic numbers because irrational algebraic numbers are expressible in terms of areas of suitable domains.
In the other direction,
Q
{\displaystyle Q}
can be restricted to be the constant function
1
{\displaystyle 1}
or
−
1
{\displaystyle -1}
, by replacing the integrand with an integral of
±
1
{\displaystyle \pm 1}
over a region defined by a polynomial in additional variables.
In other words, a (nonnegative) period is the volume of a region in
R
n
{\displaystyle \mathbb {R} ^{n}}
defined by polynomial inequalities with rational coefficients.
Properties and motivation
Periods are intended to bridge the gap between the countable algebraic numbers, whose properties have been well-studied, and transcendental numbers, which are uncountable and hard to describe apart from a very few specific examples. The latter are also not generally computable. Periods include some of those transcendental numbers that can be described in an algorithmic way and only contain a finite amount of information.
The ring of periods
P
{\displaystyle {\mathcal {P}}}
lies in between the fields of algebraic numbers
Q
¯
{\displaystyle \mathbb {\overline {Q}} }
and complex numbers
C
{\displaystyle \mathbb {C} }
and is countable. The periods themselves are all computable, and in particular definable.
Numbers known to be periods
The following numbers are among the ones known to be periods:
Open questions
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".
They 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 (in both the integrand and the domain), changes of variables, and the fundamental theorem of calculus
∫
a
b
f
′
(
x
)
d
x
=
f
(
b
)
−
f
(
a
)
{\displaystyle \int _{a}^{b}f'(x)\,dx=f(b)-f(a)}
(or, more generally, Stokes' theorem).
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 to be recursively enumerable; 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.
Extensions
The ring of periods can be extended to the ring of extended periods
P
^
{\displaystyle {\hat {\mathcal {P}}}}
by adjoining the element
1
/
π
{\displaystyle 1/\pi }
.
Permitting the integrand
Q
{\displaystyle Q}
to be the product of an algebraic function and the exponential of an algebraic function results in another extension: the exponential periods, usually denoted
E
P
{\displaystyle {\mathcal {EP}}}
. They also form a ring and are countable.
The following numbers are among the ones known to be exponential periods: