In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, specifically in
commutative algebra
Commutative algebra, first known as ideal theory, is the branch of algebra that studies commutative rings, their ideals, and modules over such rings. Both algebraic geometry and algebraic number theory build on commutative algebra. Prominent ...
, the power sum symmetric polynomials are a type of basic building block for
symmetric polynomial
In mathematics, a symmetric polynomial is a polynomial in variables, such that if any of the variables are interchanged, one obtains the same polynomial. Formally, is a ''symmetric polynomial'' if for any permutation of the subscripts one has ...
s, in the sense that every symmetric polynomial with
rational
Rationality is the quality of being guided by or based on reasons. In this regard, a person acts rationally if they have a good reason for what they do or a belief is rational if it is based on strong evidence. This quality can apply to an abi ...
coefficient
In mathematics, a coefficient is a multiplicative factor in some term of a polynomial, a series, or an expression; it is usually a number, but may be any expression (including variables such as , and ). When the coefficients are themselves var ...
s can be expressed as a sum and difference of products of power sum symmetric polynomials with rational coefficients. However, not every symmetric polynomial with
integral
In mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented i ...
coefficients is generated by integral combinations of products of power-sum polynomials: they are a generating set over the ''rationals,'' but not over the ''integers.''
Definition
The power sum symmetric polynomial of
degree
Degree may refer to:
As a unit of measurement
* Degree (angle), a unit of angle measurement
** Degree of geographical latitude
** Degree of geographical longitude
* Degree symbol (°), a notation used in science, engineering, and mathematics
...
''k'' in
variables ''x''
1, ..., ''x''
''n'', written ''p''
''k'' for ''k'' = 0, 1, 2, ..., is the sum of all ''k''th
powers of the variables. Formally,
:
The first few of these polynomials are
:
:
:
:
Thus, for each nonnegative integer
, there exists exactly one power sum symmetric polynomial of degree
in
variables.
The
polynomial ring
In mathematics, especially in the field of algebra, a polynomial ring or polynomial algebra is a ring (which is also a commutative algebra) formed from the set of polynomials in one or more indeterminates (traditionally also called variables) ...
formed by taking all
integral
In mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented i ...
linear combinations of products of the power sum symmetric polynomials is a
commutative ring
In mathematics, a commutative ring is a ring in which the multiplication operation is commutative. The study of commutative rings is called commutative algebra. Complementarily, noncommutative algebra is the study of ring properties that are not sp ...
.
Examples
The following lists the
power sum symmetric polynomials of positive degrees up to ''n'' for the first three positive values of
In every case,
is one of the polynomials. The list goes up to degree ''n'' because the power sum symmetric polynomials of degrees 1 to ''n'' are basic in the sense of the theorem stated below.
For ''n'' = 1:
:
For ''n'' = 2:
:
:
For ''n'' = 3:
:
:
:
Properties
The set of power sum symmetric polynomials of degrees 1, 2, ..., ''n'' in ''n'' variables
generates the
ring
Ring may refer to:
* Ring (jewellery), a round band, usually made of metal, worn as ornamental jewelry
* To make a sound with a bell, and the sound made by a bell
:(hence) to initiate a telephone connection
Arts, entertainment and media Film and ...
of
symmetric polynomial
In mathematics, a symmetric polynomial is a polynomial in variables, such that if any of the variables are interchanged, one obtains the same polynomial. Formally, is a ''symmetric polynomial'' if for any permutation of the subscripts one has ...
s in ''n'' variables. More specifically:
:Theorem. The ring of symmetric polynomials with rational coefficients equals the rational polynomial ring
The same is true if the coefficients are taken in any
field
Field may refer to:
Expanses of open ground
* Field (agriculture), an area of land used for agricultural purposes
* Airfield, an aerodrome that lacks the infrastructure of an airport
* Battlefield
* Lawn, an area of mowed grass
* Meadow, a grass ...
of
characteristic 0.
However, this is not true if the coefficients must be integers. For example, for ''n'' = 2, the symmetric polynomial
:
has the expression
:
which involves fractions. According to the theorem this is the only way to represent
in terms of ''p''
1 and ''p''
2. Therefore, ''P'' does not belong to the integral polynomial ring
For another example, the
elementary symmetric polynomial
In mathematics, specifically in commutative algebra, the elementary symmetric polynomials are one type of basic building block for symmetric polynomials, in the sense that any symmetric polynomial can be expressed as a polynomial in elementary sy ...
s ''e''
''k'', expressed as polynomials in the power sum polynomials, do not all have integral coefficients. For instance,
:
The theorem is also untrue if the field has characteristic different from 0. For example, if the field ''F'' has characteristic 2, then
, so ''p''
1 and ''p''
2 cannot generate ''e''
2 = ''x''
1''x''
2.
''Sketch of a partial proof of the theorem'': By
Newton's identities
In mathematics, Newton's identities, also known as the Girard–Newton formulae, give relations between two types of symmetric polynomials, namely between power sums and elementary symmetric polynomials. Evaluated at the roots of a monic polynomia ...
the power sums are functions of the elementary symmetric polynomials; this is implied by the following
recurrence relation
In mathematics, a recurrence relation is an equation according to which the nth term of a sequence of numbers is equal to some combination of the previous terms. Often, only k previous terms of the sequence appear in the equation, for a parameter ...
, though the explicit function that gives the power sums in terms of the ''e''
''j'' is complicated:
:
Rewriting the same recurrence, one has the elementary symmetric polynomials in terms of the power sums (also implicitly, the explicit formula being complicated):
:
This implies that the elementary polynomials are rational, though not integral, linear combinations of the power sum polynomials of degrees 1, ..., ''n''.
Since the elementary symmetric polynomials are an algebraic basis for all symmetric polynomials with coefficients in a field, it follows that every symmetric polynomial in ''n'' variables is a polynomial function
of the power sum symmetric polynomials ''p''
1, ..., ''p''
''n''. That is, the ring of symmetric polynomials is contained in the ring generated by the power sums,
Because every power sum polynomial is symmetric, the two rings are equal.
(This does not show how to prove the polynomial ''f'' is unique.)
For another system of symmetric polynomials with similar properties see
complete homogeneous symmetric polynomial
In mathematics, specifically in algebraic combinatorics and commutative algebra, the complete homogeneous symmetric polynomials are a specific kind of symmetric polynomials. Every symmetric polynomial can be expressed as a polynomial expression ...
s.
See also
*
Representation theory
Representation theory is a branch of mathematics that studies abstract algebraic structures by ''representing'' their elements as linear transformations of vector spaces, and studies modules over these abstract algebraic structures. In essen ...
*
Newton's identities
In mathematics, Newton's identities, also known as the Girard–Newton formulae, give relations between two types of symmetric polynomials, namely between power sums and elementary symmetric polynomials. Evaluated at the roots of a monic polynomia ...
References
*
Ian G. Macdonald (1979), ''Symmetric Functions and Hall Polynomials''. Oxford Mathematical Monographs. Oxford: Clarendon Press.
* Ian G. Macdonald (1995), ''Symmetric Functions and Hall Polynomials'', second ed. Oxford: Clarendon Press. (paperback, 1998).
*
Richard P. Stanley
Richard Peter Stanley (born June 23, 1944) is an Emeritus Professor of Mathematics at the Massachusetts Institute of Technology, in Cambridge, Massachusetts. From 2000 to 2010, he was the Norman Levinson Professor of Applied Mathematics. He r ...
(1999), ''Enumerative Combinatorics'', Vol. 2. Cambridge: Cambridge University Press. {{ISBN, 0-521-56069-1
Homogeneous polynomials
Symmetric functions