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A polynomial <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091700/s0917001.png" />, with coefficients in a field or a commutative associative ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091700/s0917002.png" /> with a unit, which is a [[Symmetric function|symmetric function]] in its variables, that is, is invariant under all permutations of the variables:
+
{{TEX|done}}
 +
A polynomial $f$ with coefficients in a field or a commutative associative ring $K$ with a unit, which is a [[Symmetric function|symmetric function]] in its variables, that is, is invariant under all permutations of the variables:
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091700/s0917003.png" /></td> <td valign="top" style="width:5%;text-align:right;">(*)</td></tr></table>
+
\[
 +
\label{symm}
 +
f(x_1,\ldots,x_n) = f(\pi(x_1),\ldots,\pi(x_n)).
 +
\]
  
The symmetric polynomials form the algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091700/s0917004.png" /> over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091700/s0917005.png" />.
+
The symmetric polynomials form the algebra $S(x_1,\ldots,x_n)$ over $K$.
  
The most important examples of symmetric polynomials are the elementary symmetric polynomials
+
The most important examples of symmetric polynomials are the ''[[elementary symmetric polynomial]]s''
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091700/s0917006.png" /></td> </tr></table>
+
\[
 +
s_k(x_1,\ldots,x_n) = \sum_{1 \leq i_1 < \ldots < i_k \leq n} x_{i_1} \ldots x_{i_k}
 +
\]
  
 
and the power sums
 
and the power sums
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091700/s0917007.png" /></td> </tr></table>
+
\[
 +
p_k(x_1,\ldots,x_n) = x_1^k + \ldots + x_n^k.
 +
\]
  
 
The latter can be expressed in terms of elementary symmetric polynomials by recurrence formulas, called Newton's formulas:
 
The latter can be expressed in terms of elementary symmetric polynomials by recurrence formulas, called Newton's formulas:
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091700/s0917008.png" /></td> </tr></table>
+
\[
 +
\begin{aligned}
 +
p_k - p_{k-1} s_1 + p_{k-2} s_2 + \ldots  + (-1)^{k-1} p_1 s_{k-1}+  (-1)^{k} k s_{k} &= 0 \quad &\text{if }1 \leq k \leq n,
 +
\\
 +
p_k - p_{k-1} s_1 + p_{k-2} s_2 + \ldots  + (-1)^{n-1} p_{k-n+1} s_{n-1}+  (-1)^{n} p_{k-n} s_{n} &= 0 \quad &\text{if }k > n.
 +
\end{aligned}
 +
\]
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091700/s0917009.png" /></td> </tr></table>
+
For the elementary symmetric polynomials $s_1,\ldots,s_k$ ($1 \leq k \leq n$) of the roots of an arbitrary polynomial in one variable with leading coefficient 1, $x^n+\alpha_1 x^{n-1}+ \ldots  + \alpha_n$, one has $\alpha_k=(-1)^k s_k$ (see [[Viète theorem|Viète theorem]]).
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091700/s09170010.png" /></td> </tr></table>
+
The fundamental theorem on symmetric polynomials: Every symmetric polynomial is a polynomial in the elementary symmetric polynomials, and this representation is unique. In other words, the elementary symmetric polynomials are a set of free generators for the algebra $S(x_1,\ldots,x_n)$. If the field has characteristic 0, then the polynomials $p_1,\ldots,p_n$ also form a set of free generators of this algebra.
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091700/s09170011.png" /></td> </tr></table>
+
A skew-symmetric, or alternating, polynomial is a polynomial $f(x_1,\ldots,x_n)$ satisfying the relation \ref{symm} if $\pi$ is even and the relation
  
For the elementary symmetric polynomials <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091700/s09170012.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091700/s09170013.png" />) of the roots of an arbitrary polynomial in one variable with leading coefficient 1, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091700/s09170014.png" />, one has <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091700/s09170015.png" /> (see [[Viète theorem|Viète theorem]]).
+
\[
 +
f(x_1,\ldots,x_n) = -f(\pi(x_1),\ldots,\pi(x_n))
 +
\]
  
The fundamental theorem on symmetric polynomials: Every symmetric polynomial is a polynomial in the elementary symmetric polynomials, and this representation is unique. In other words, the elementary symmetric polynomials are a set of free generators for the algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091700/s09170016.png" />. If the field has characteristic 0, then the polynomials <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091700/s09170017.png" /> also form a set of free generators of this algebra.
+
if $\pi$ is odd. Any skew-symmetric polynomial can be written {{Cite|Pr|Thm. 3.1.2}} in the form $\Delta_n g$, where $g$ is a symmetric polynomial and
  
A skew-symmetric, or alternating, polynomial is a polynomial <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091700/s09170018.png" /> satisfying the relation (*) if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091700/s09170019.png" /> is even and the relation
+
\[
 +
\Delta_n = \prod_{i<j} (x_i-x_j).
 +
\]
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091700/s09170020.png" /></td> </tr></table>
+
This representation is not unique, in view of the relation $\Delta_n^2=\operatorname{Dis}(s_1,\ldots,s_n)$.
 
 
if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091700/s09170021.png" /> is odd. Any skew-symmetric polynomial can be written in the form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091700/s09170022.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091700/s09170023.png" /> is a symmetric polynomial and
 
 
 
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091700/s09170024.png" /></td> </tr></table>
 
 
 
This representation is not unique, in view of the relation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091700/s09170025.png" />.
 
  
 
====References====
 
====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> A.G. Kurosh,   "Higher algebra" , MIR  (1972)  (Translated from Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> A.I. Kostrikin,   "Introduction to algebra" , Springer  (1982)  (Translated from Russian)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top"> A.P. Mishina,   I.V. Proskuryakov,   "Higher algebra. Linear algebra, polynomials, general algebra" , Pergamon (1965)  (Translated from Russian)</TD></TR></table>
+
{|
 
+
||{{Ref|Ku}}|| A.G. Kurosh, "Higher algebra", MIR  (1972)  (Translated from Russian)
 
+
|-
 +
||{{Ref|Ko}}|| A.I. Kostrikin, "Introduction to algebra", Springer  (1982)  (Translated from Russian)
 +
|-
 +
||{{Ref|MP}}|| A.P. Mishina, I.V. Proskuryakov, "Higher algebra. Linear algebra, polynomials, general algebra", Pergamon (1965)  (Translated from Russian)
 +
|-
 +
||{{Ref|Pr}}|| V. V. Prasolov, "Polynomials", vol. 11 of "Algorithms and Computation in Mathematics", Springer (2004).
 +
|}
  
 
====Comments====
 
====Comments====
Another important set of symmetric polynomials, which appear in the representations of the symmetric group, are the Schur polynomials (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091700/s09170027.png" />-functions) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091700/s09170028.png" />. These are defined for any partition <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091700/s09170029.png" />, and include as special cases the above functions, e.g. <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091700/s09170030.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091700/s09170031.png" /> (see, e.g., [[#References|[a4]]], Chapt. VI).
+
Another important set of symmetric polynomials, which appear in the representations of the symmetric group, are the Schur polynomials ($S$-functions) $S_{\{\lambda\}}$. These are defined for any partition $\{\lambda_1,\ldots,\lambda_p\}=\lambda$, and include as special cases the above functions, e.g. $S_{\{1,\ldots,1\}}=s_k$, $S_{\{ k \}}=p_k$ (see, e.g., {{Cite|Li|Chapt. VI}}).
  
In general, the discriminant of the polynomial <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091700/s09170032.png" /> with roots <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091700/s09170033.png" /> is defined as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091700/s09170034.png" />, and satisfies
+
In general, the discriminant of the polynomial $a_0 x^n + \ldots + a_n $  with roots $x_1,\ldots,x_n$ is defined as $D=a_0^{2n-2} \prod_{1 \leq i < j \leq n} (x_i-x_j)^2$, and satisfies
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091700/s09170035.png" /></td> </tr></table>
+
\[
 +
D = a_0^{2n-2} \begin{vmatrix}
 +
p_0 & \ldots & p_{n-1}
 +
\\
 +
p_1 &\ldots & p_n
 +
\\
 +
\ldots & \ldots & \ldots
 +
\\
 +
p_{n-1} & \ldots & p_{2n-2}
 +
\end{vmatrix},
 +
\]
  
with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091700/s09170036.png" />.
+
with $p_0(x_1,\ldots,x_n)=n$.
  
 
See [[Discriminant|Discriminant]].
 
See [[Discriminant|Discriminant]].
  
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091700/s09170037.png" /> be the [[Alternating group|alternating group]], consisting of the even permutations. The ring of polynomials <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091700/s09170038.png" /> of polynomials over a field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091700/s09170039.png" /> obviously contains the elementary symmetric functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091700/s09170040.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091700/s09170041.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091700/s09170042.png" /> is not of characteristic <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091700/s09170043.png" />, the ring of polynomials is generated by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091700/s09170044.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091700/s09170045.png" />, and the ideal of relations is generated by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091700/s09170046.png" />. The condition <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091700/s09170047.png" /> is also necessary for the statement that every skew-symmetric polynomial is of the form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091700/s09170048.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091700/s09170049.png" /> symmetric. More precisely, what is needed for this is that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091700/s09170050.png" /> implies <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091700/s09170051.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091700/s09170052.png" />.
+
Let $A_n \subset S_n$ be the [[Alternating group|alternating group]], consisting of the even permutations. The ring of polynomials $k[x_1,\ldots,x_n]^{A_n}$ of polynomials over a field $k$ obviously contains the elementary symmetric functions $s_1,\ldots,s_n$ and $\Delta_n=\prod_{i<j} (x_i-x_j)$. If $k$ is not of characteristic $2$, the ring of polynomials is generated by $s_1,\ldots,s_n$ and $\Delta$, and the ideal of relations is generated by $\Delta_n^2=\operatorname{Dis}(s_1,\ldots,s_n)$. The condition $\operatorname{char} k \neq 2$ is also necessary for the statement that every skew-symmetric polynomial is of the form $\Delta_n g$  with $g$ symmetric. More precisely, what is needed for this is that $2u=0$ implies $u=0$ for $u \in k$.
  
 
====References====
 
====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> N. Jacobson,  "Basic algebra" , '''1''' , Freeman  (1974)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  A.G. Kurosh,  "An introduction to algebra" , MIR  (1971)  (Translated from Russian)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  B.L. van der Waerden,  "Algebra" , '''1''' , Springer  (1967)  (Translated from German)</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  D.E. Littlewood,  "The theory of group characters and matrix representations of groups" , Clarendon Press  (1950)</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  V. Poénaru,   "Singularités <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091700/s09170053.png" /> en présence de symmétrie" , Springer  (1976)  pp. 14ff</TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top">  P.M. Cohn,  "Algebra" , '''1''' , Wiley  (1982)  pp. 181</TD></TR></table>
+
{|
 +
||{{Ref|Ja}}|| N. Jacobson,  "Basic algebra" , '''1''', Freeman  (1974)
 +
|-
 +
||{{Ref|Ku2}}|| A.G. Kurosh,  "An introduction to algebra" , MIR  (1971)  (Translated from Russian)
 +
|-
 +
||{{Ref|vdW}}|| B.L. van der Waerden,  "Algebra" , '''1''', Springer  (1967)  (Translated from German)
 +
|-
 +
||{{Ref|Li}}|| D.E. Littlewood,  "The theory of group characters and matrix representations of groups" , Clarendon Press  (1950)
 +
|-
 +
||{{Ref|Po}}||V. Poénaru, "Singularités $\mathcal{C}^\infty$ en présence de symmétrie" , Springer  (1976), pp. 14ff
 +
|-
 +
||{{Ref|Co}}||P.M. Cohn,  "Algebra" , '''1''', Wiley  (1982)  pp. 181
 +
|}

Latest revision as of 20:34, 13 September 2016

A polynomial $f$ with coefficients in a field or a commutative associative ring $K$ with a unit, which is a symmetric function in its variables, that is, is invariant under all permutations of the variables:

\[ \label{symm} f(x_1,\ldots,x_n) = f(\pi(x_1),\ldots,\pi(x_n)). \]

The symmetric polynomials form the algebra $S(x_1,\ldots,x_n)$ over $K$.

The most important examples of symmetric polynomials are the elementary symmetric polynomials

\[ s_k(x_1,\ldots,x_n) = \sum_{1 \leq i_1 < \ldots < i_k \leq n} x_{i_1} \ldots x_{i_k} \]

and the power sums

\[ p_k(x_1,\ldots,x_n) = x_1^k + \ldots + x_n^k. \]

The latter can be expressed in terms of elementary symmetric polynomials by recurrence formulas, called Newton's formulas:

\[ \begin{aligned} p_k - p_{k-1} s_1 + p_{k-2} s_2 + \ldots + (-1)^{k-1} p_1 s_{k-1}+ (-1)^{k} k s_{k} &= 0 \quad &\text{if }1 \leq k \leq n, \\ p_k - p_{k-1} s_1 + p_{k-2} s_2 + \ldots + (-1)^{n-1} p_{k-n+1} s_{n-1}+ (-1)^{n} p_{k-n} s_{n} &= 0 \quad &\text{if }k > n. \end{aligned} \]

For the elementary symmetric polynomials $s_1,\ldots,s_k$ ($1 \leq k \leq n$) of the roots of an arbitrary polynomial in one variable with leading coefficient 1, $x^n+\alpha_1 x^{n-1}+ \ldots + \alpha_n$, one has $\alpha_k=(-1)^k s_k$ (see Viète theorem).

The fundamental theorem on symmetric polynomials: Every symmetric polynomial is a polynomial in the elementary symmetric polynomials, and this representation is unique. In other words, the elementary symmetric polynomials are a set of free generators for the algebra $S(x_1,\ldots,x_n)$. If the field has characteristic 0, then the polynomials $p_1,\ldots,p_n$ also form a set of free generators of this algebra.

A skew-symmetric, or alternating, polynomial is a polynomial $f(x_1,\ldots,x_n)$ satisfying the relation \ref{symm} if $\pi$ is even and the relation

\[ f(x_1,\ldots,x_n) = -f(\pi(x_1),\ldots,\pi(x_n)) \]

if $\pi$ is odd. Any skew-symmetric polynomial can be written [Pr, Thm. 3.1.2] in the form $\Delta_n g$, where $g$ is a symmetric polynomial and

\[ \Delta_n = \prod_{i<j} (x_i-x_j). \]

This representation is not unique, in view of the relation $\Delta_n^2=\operatorname{Dis}(s_1,\ldots,s_n)$.

References

[Ku] A.G. Kurosh, "Higher algebra", MIR (1972) (Translated from Russian)
[Ko] A.I. Kostrikin, "Introduction to algebra", Springer (1982) (Translated from Russian)
[MP] A.P. Mishina, I.V. Proskuryakov, "Higher algebra. Linear algebra, polynomials, general algebra", Pergamon (1965) (Translated from Russian)
[Pr] V. V. Prasolov, "Polynomials", vol. 11 of "Algorithms and Computation in Mathematics", Springer (2004).

Comments

Another important set of symmetric polynomials, which appear in the representations of the symmetric group, are the Schur polynomials ($S$-functions) $S_{\{\lambda\}}$. These are defined for any partition $\{\lambda_1,\ldots,\lambda_p\}=\lambda$, and include as special cases the above functions, e.g. $S_{\{1,\ldots,1\}}=s_k$, $S_{\{ k \}}=p_k$ (see, e.g., [Li, Chapt. VI]).

In general, the discriminant of the polynomial $a_0 x^n + \ldots + a_n $ with roots $x_1,\ldots,x_n$ is defined as $D=a_0^{2n-2} \prod_{1 \leq i < j \leq n} (x_i-x_j)^2$, and satisfies

\[ D = a_0^{2n-2} \begin{vmatrix} p_0 & \ldots & p_{n-1} \\ p_1 &\ldots & p_n \\ \ldots & \ldots & \ldots \\ p_{n-1} & \ldots & p_{2n-2} \end{vmatrix}, \]

with $p_0(x_1,\ldots,x_n)=n$.

See Discriminant.

Let $A_n \subset S_n$ be the alternating group, consisting of the even permutations. The ring of polynomials $k[x_1,\ldots,x_n]^{A_n}$ of polynomials over a field $k$ obviously contains the elementary symmetric functions $s_1,\ldots,s_n$ and $\Delta_n=\prod_{i<j} (x_i-x_j)$. If $k$ is not of characteristic $2$, the ring of polynomials is generated by $s_1,\ldots,s_n$ and $\Delta$, and the ideal of relations is generated by $\Delta_n^2=\operatorname{Dis}(s_1,\ldots,s_n)$. The condition $\operatorname{char} k \neq 2$ is also necessary for the statement that every skew-symmetric polynomial is of the form $\Delta_n g$ with $g$ symmetric. More precisely, what is needed for this is that $2u=0$ implies $u=0$ for $u \in k$.

References

[Ja] N. Jacobson, "Basic algebra" , 1, Freeman (1974)
[Ku2] A.G. Kurosh, "An introduction to algebra" , MIR (1971) (Translated from Russian)
[vdW] B.L. van der Waerden, "Algebra" , 1, Springer (1967) (Translated from German)
[Li] D.E. Littlewood, "The theory of group characters and matrix representations of groups" , Clarendon Press (1950)
[Po] V. Poénaru, "Singularités $\mathcal{C}^\infty$ en présence de symmétrie" , Springer (1976), pp. 14ff
[Co] P.M. Cohn, "Algebra" , 1, Wiley (1982) pp. 181
How to Cite This Entry:
Symmetric polynomial. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Symmetric_polynomial&oldid=16090
This article was adapted from an original article by O.A. Ivanova (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article