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 ${\displaystyle {\mathcal {P}}}$ 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]

Definition

A number ${\displaystyle \alpha }$ is a period if it can be expressed as an integral of the form

${\displaystyle \alpha =\int _{P(x_{1},\ldots ,x_{n})\geq 0}Q(x_{1},\ldots ,x_{n})\ \mathrm {d} x_{1}\ldots \mathrm {d} x_{n}}$

where ${\displaystyle P}$ is a polynomial and ${\displaystyle Q}$ a rational function on ${\displaystyle \mathbb {R} ^{n}}$ with rational coefficients. ^[1] A complex number is a period if its real and imaginary parts are periods.

An alternative definition allows ${\displaystyle P}$ and ${\displaystyle Q}$ to be algebraic functions; this looks more general, but is equivalent. The coefficients of the rational functions and polynomials can also be generalised to algebraic numbers because irrational algebraic numbers are expressible in terms of areas of suitable domains.

In the other direction, ${\displaystyle Q}$ can be restricted to be the constant function ${\displaystyle 1}$ or ${\displaystyle -1}$, by replacing the integrand with an integral of ${\displaystyle \pm 1}$ over a region defined by a polynomial in additional variables.

In other words, a (nonnegative) period is the volume of a region in ${\displaystyle \mathbb {R} ^{n}}$ defined by polynomial inequalities with rational coefficients. ^{[2] [4]}

Properties and motivation

The periods are intended to bridge the gap between the well-behaved algebraic numbers, which form a class too narrow to include many common mathematical constants and the transcendental numbers, which are uncountable and apart from very few specific examples hard to describe. They are also not generally computable.

The ring of periods ${\displaystyle {\mathcal {P}}}$ lies in between the fields of algebraic numbers ${\displaystyle \mathbb {\overline {Q}} }$ and complex numbers ${\displaystyle \mathbb {C} }$ and is countable. ^[5] The periods themselves are all computable, ^[6] and in particular definable. It is: ${\displaystyle \mathbb {\overline {Q}} \subset {\mathcal {P}}\subset \mathbb {C} }$.

Periods include some of those transcendental numbers, that can be described in an algorithmic way and only contain a finite amount of information. ^[2]

Numbers known to be periods

The following numbers are among the ones known to be periods: ^{[1] [2] [4] [7]}

Number	Example of period integral
Any algebraic number ${\displaystyle \alpha }$.	${\displaystyle \alpha =\int _{0}^{\alpha }\mathrm {d} x}$
The natural logarithm of any positive algebraic number ${\displaystyle \alpha >0}$.	${\displaystyle \ln(\alpha )=\int _{1}^{\alpha }{\frac {1}{x}}\ \mathrm {d} x}$
The inverse trigonometric functions at algebraic numbers ${\displaystyle \alpha }$ in their domain.	${\displaystyle \arctan(\alpha )=\int _{0}^{\alpha }{\frac {1}{1+x^{2}}}\ \mathrm {d} x}$
The inverse hyperbolic functions at algebraic numbers ${\displaystyle \alpha }$ in their domain.	${\displaystyle {\text{artanh}}(\alpha )=\int _{0}^{\alpha }{\frac {1}{1-x^{2}}}\ \mathrm {d} x}$
The number ${\displaystyle \pi }$.	${\displaystyle \pi =\int _{0}^{1}{\frac {4}{x^{2}+1}}\ \mathrm {d} x}$
Integer values of the Riemann zeta function ${\displaystyle \zeta (s)}$ for ${\displaystyle s\geq 2}$ as well as several multiple zeta values. In particular: Even powers ${\displaystyle \pi ^{2n}}$ and Apéry's constant ${\displaystyle \zeta (3)}$.	${\displaystyle \zeta (3)=\int _{0}^{1}\int _{0}^{1}\int _{0}^{1}{\frac {\mathrm {d} x\mathrm {d} y\mathrm {d} z}{1-xyz}}}$
Integer values of the Dirichlet beta function ${\displaystyle \beta (s)}$. In particular: Odd powers ${\displaystyle \pi ^{2n+1}}$ and Catalan's constant ${\displaystyle G}$.	${\displaystyle G=\int _{0}^{1}\int _{0}^{1}{\frac {\mathrm {d} x\mathrm {d} y}{1+x^{2}y^{2}}}}$
Certain values of the Clausen function ${\displaystyle {\text{Cl}}_{2}(z)}$ at rational multiples of ${\displaystyle \pi }$. In particular: The Gieseking constant ${\displaystyle {\text{Cl}}_{2}({\tfrac {1}{3}}\pi )}$.	${\displaystyle {\text{Cl}}_{2}({\tfrac {1}{3}}\pi )=2\int _{0}^{1}\int _{1}^{1+y}{\frac {\mathrm {d} x\mathrm {d} y}{x{\sqrt {(1-y)(3+y)}}}}}$
Rational values of the polygamma function ${\displaystyle \psi _{m}(z)}$ for ${\displaystyle z\in \mathbb {Q} ^{+}}$ in its domain and ${\displaystyle m\in \mathbb {Z} ^{+}}$.	${\displaystyle \psi _{m}(z)=-\int _{0}^{1}{\frac {y^{z-1}}{1-y}}\left[\int _{1}^{y}{\frac {\mathrm {d} x}{x}}\right]^{m}\,\mathrm {d} y}$
The polylogarithm ${\displaystyle {\text{Li}}_{s}(\alpha )}$ at algebraic numbers ${\displaystyle \alpha }$ in its domain and ${\displaystyle s\in \mathbb {Z} ^{+}}$.	${\displaystyle {\text{Li}}_{2}(\alpha )=-\int _{0}^{\alpha }\int _{1}^{1-y}{\frac {1}{xy}}\ \mathrm {d} x\mathrm {d} y}$
The inverse tangent integral ${\displaystyle {\text{Ti}}_{n}(\alpha )}$ at algebraic numbers ${\displaystyle \alpha }$ in its domain and ${\displaystyle n\in \mathbb {Z} ^{+}}$.	${\displaystyle {\text{Ti}}_{2}(\alpha )=\int _{0}^{\alpha }\int _{0}^{y}{\frac {1}{y(1+x^{2})}}\ \mathrm {d} x\mathrm {d} y}$
Values of elliptic integrals with algebraic bounds. In particular: The perimeter ${\displaystyle P}$ of an ellipse with algebraic radii ${\displaystyle a}$ and ${\displaystyle b}$.	${\displaystyle P=\int _{-b}^{b}{\sqrt {1+{\frac {a^{2}x^{2}}{b^{4}-b^{2}x^{2}}}}}\mathrm {d} x}$
Several numbers related to the gamma and beta functions, such as values ${\displaystyle \Gamma (p/q)^{q}}$ for ${\displaystyle p,q\in \mathbb {Z} ^{+}}$ and ${\displaystyle \mathrm {B} ({\tfrac {1}{n}},{\tfrac {1}{n}})}$ for ${\displaystyle n\in \mathbb {Z} ^{+}}$. In particular: The lemniscate constant ${\displaystyle \varpi }$.	${\displaystyle \mathrm {B} \left({\tfrac {1}{n}},{\tfrac {1}{n}}\right)=2n\int _{0}^{1}{\sqrt[{n}]{1-x^{n}}}\ \mathrm {d} x}$
Special values of hypergeometric functions at algebraic arguments.	${\displaystyle _{2}F_{1}(-{\tfrac {1}{2}},{\tfrac {1}{3}};{\tfrac {4}{3}};-1)={\frac {1}{3}}\int _{0}^{1}{\frac {\sqrt {1+x}}{x^{2/3}}}\mathrm {d} x}$
Special values of modular forms at certain arguments.	[2]
Sums and products of 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".

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 (in both the integrand and the domain), changes of variables, and the Newton–Leibniz formula

${\displaystyle \int _{a}^{b}f'(x)\,dx=f(b)-f(a)}$

(or, more generally, the Stokes 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.

Further open questions consist of proving which known mathematical constants do not belong to the ring of 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. It is also possible to construct artificial examples of computable numbers which are not periods. ^[8] However there are no computable numbers proven not to be periods, which have not been artificially constructed for that purpose.

It is conjectured that 1/π, Euler's number e and the Euler–Mascheroni constant γ are not periods. ^[2]

Kontsevich and Zagier suspect these problems to be very hard and remain open a long time.

Extensions

The ring of periods can be widened to the ring of extended periods${\displaystyle {\hat {\mathcal {P}}}}$ by adjoining the element 1/π. ^[2]

Permitting the integrand ${\displaystyle Q}$ to be the product of an algebraic function and the exponential of an algebraic function, results in another extension: the exponential periods${\displaystyle {\mathcal {E}}{\mathcal {P}}}$. ^{[2] [4] [9]} They also form a ring and are countable. It is ${\displaystyle {\overline {\mathbb {Q} }}\subset {\mathcal {P}}\subseteq {\mathcal {EP}}\subset \mathbb {C} }$.

The following numbers are among the ones known to be exponential periods: ^{[2] [4] [10]}

Number	Example of exponential period integral
Any algebraic period ${\displaystyle I\in {\mathcal {P}}}$
Numbers of the form ${\displaystyle e^{\alpha }}$ with ${\displaystyle \alpha \in {\overline {\mathbb {Q} }}}$. In particular: The number ${\displaystyle e}$.	${\displaystyle e^{\alpha }=\int _{-\infty }^{\alpha }e^{x}\mathrm {d} x}$
The functions ${\displaystyle \sin(\alpha )}$ and ${\displaystyle \cos(\alpha )}$ at algebraic values.	${\displaystyle \sin(\alpha )={\frac {1}{2}}\int _{-\alpha }^{\alpha }e^{ix}\mathrm {d} x}$
The functions ${\displaystyle \sinh(\alpha )}$ and ${\displaystyle \cosh(\alpha )}$ at algebraic values.	${\displaystyle \sinh(\alpha )={\frac {1}{2}}\int _{-\alpha }^{\alpha }e^{x}\mathrm {d} x}$
Rational values of the gamma function ${\displaystyle \Gamma (p/q)}$ with ${\displaystyle p,q\in \mathbb {Z} ^{+}}$. In particular: ${\displaystyle {\sqrt {\pi }}}$.	${\displaystyle \Gamma (p/q)=\int _{0}^{\infty }x^{{\frac {p}{q}}-1}e^{-x}\mathrm {d} x}$
Euler's constant ${\displaystyle \gamma }$ and positive rational values of the digamma function ${\displaystyle \psi _{0}(p/q)}$. [11]	${\displaystyle \gamma =-\int _{0}^{\infty }\int _{1}^{y}{\frac {e^{-y}}{x}}\ \mathrm {d} x\mathrm {d} y}$
Algebraic values of the exponential integral and the Gompertz constant ${\displaystyle \delta }$.	${\displaystyle \delta =\int _{0}^{\infty }{\frac {1}{1+x}}e^{-x}\mathrm {d} x}$
Algebraic values of several trigonometric integrals.	${\displaystyle {\text{Si}}(\alpha )=\int _{0}^{\alpha }\int _{-y}^{y}{\frac {e^{ix}}{2y}}\mathrm {d} x\mathrm {d} y}$
Certain values of Bessel functions.	[2]
Sums and products of exponential periods.

See also

• Transcendental number theory
• Mathematical constant
• L-function
• Jacobian variety
• Gauss–Manin connection
• Mixed motives (math)
• Tannakian formalism

Number systems Complex ${\displaystyle$ :\;\mathbb {C} } Real ${\displaystyle$ :\;\mathbb {R} } Rational ${\displaystyle$ :\;\mathbb {Q} } Integer ${\displaystyle$ :\;\mathbb {Z} } Natural ${\displaystyle$ :\;\mathbb {N} } Zero: 0 One: 1 Prime numbers Composite numbers Negative integers Fraction Finite decimal Dyadic (finite binary) Repeating decimal Irrational Algebraic irrational Irrational period Transcendental Imaginary

References

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].
4. Lagarias, Jeffrey C. (2013-07-19). "Euler's constant: Euler's work and modern developments". Bulletin of the American Mathematical Society. 50 (4): 527–628. arXiv: 1303.1856 . doi:10.1090/S0273-0979-2013-01423-X. ISSN   0273-0979.
5. Müller-Stach, Stefan (2014-07-09), What is a period ?, arXiv: 1407.2388
6. Tent, Katrin; Ziegler, Martin (2010). "Computable functions of the reals" (PDF). Münster Journal of Mathematics. 3: 43–66.
7. Waldschmidt, Michel (2006). "Transcendence of periods: the state of the art". Pure and Applied Mathematics Quarterly. 2 (2): 435–463. doi:10.4310/PAMQ.2006.v2.n2.a3.
8. Yoshinaga, Masahiko (2008-05-03). "Periods and elementary real numbers". arXiv: 0805.0349 [math.AG].
9. Commelin, Johan; Habegger, Philipp; Huber, Annette (2022-03-30). "Exponential periods and o-minimality". arXiv: 2007.08280 [math.NT].
10. Belkale, Prakash; Brosnan, Patrick (2003). "Periods and Igusa local zeta functions". International Mathematics Research Notices. 2003 (49): 2655. doi: 10.1155/S107379280313142X . Retrieved 2024-09-21.
11. Using the following integral representation ${\displaystyle \psi (z)=-\gamma +\int _{0}^{1}\left({\frac {1-t^{z-1}}{1-t}}\right)\,dt}$ for positive z and the exponential period integral of ${\displaystyle \gamma }$ one obtains all positive rational digamma values as a sum of two exponential period integrals.

External links

• PlanetMath: Period