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| In [[mathematics]], specifically in [[commutative algebra]], the '''power sum symmetric polynomials''' are a type of basic building block for [[symmetric polynomial]]s, in the sense that every symmetric polynomial with rational coefficients 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 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.''
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| ==Definition==
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| The power sum symmetric polynomial of degree ''k'' in <math>n</math> variables ''x''<sub>1</sub>, ..., ''x''<sub>''n''</sub>, written ''p''<sub>''k''</sub> for ''k'' = 0, 1, 2, ..., is the sum of all ''k''th [[Exponentiation|powers]] of the variables. Formally,
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| :<math> p_k (x_1, x_2, \dots,x_n) = \sum_{i=1}^n x_i^k \, .</math>
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| The first few of these polynomials are
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| :<math> p_0 (x_1, x_2, \dots,x_n) = n ,</math>
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| :<math> p_1 (x_1, x_2, \dots,x_n) = x_1 + x_2 + \cdots + x_n \, ,</math>
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| :<math> p_2 (x_1, x_2, \dots,x_n) = x_1^2 + x_2^2 + \cdots + x_n^2 \, ,</math>
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| :<math> p_3 (x_1, x_2, \dots,x_n) = x_1^3 + x_2^3 + \cdots + x_n^3 \, .</math>
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| Thus, for each nonnegative integer <math>k</math>, there exists exactly one power sum symmetric polynomial of degree <math>k</math> in <math>n</math> variables.
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| The [[polynomial ring]] formed by taking all integral linear combinations of products of the power sum symmetric polynomials is a [[commutative ring]].
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| ==Examples==
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| The following lists the <math>n</math> power sum symmetric polynomials of positive degrees up to ''n'' for the first three positive values of <math>n.</math> In every case, <math>p_0 = n</math> 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 Main Theorem stated below.
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| For ''n'' = 1:
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| :<math>p_1 = x_1\,.</math>
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| For ''n'' = 2:
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| :<math>p_1 = x_1 + x_2\,,</math>
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| :<math>p_2 = x_1^2 + x_2^2\,.</math>
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| For ''n'' = 3:
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| :<math>p_1 = x_1 + x_2 + x_3\,,</math>
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| :<math>p_2 = x_1^2 + x_2^2 + x_3^2\,,</math>
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| :<math>p_3 = x_1^3+x_2^3+x_3^3\,,</math>
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| ==Properties==
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| The set of power sum symmetric polynomials of degrees 1, 2, ..., ''n'' in ''n'' variables [[generator (mathematics)|generates]] the [[polynomial ring|ring]] of [[symmetric polynomial]]s in ''n'' variables. More specifically:
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| :'''Theorem'''. The ring of symmetric polynomials with rational coefficients equals the rational polynomial ring <math>\mathbb Q[p_1,\ldots,p_n].</math> The same is true if the coefficients are taken in any [[field (mathematics)|field]] whose characteristic is 0.
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| However, this is not true if the coefficients must be integers. For example, for ''n'' = 2, the symmetric polynomial
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| :<math>P(x_1,x_2) = x_1^2 x_2 + x_1 x_2^2 + x_1x_2</math>
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| has the expression
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| :<math>P(x_1,x_2) = \frac{p_1^3-p_1p_2}{2} + \frac{p_1^2-p_2}{2} \,,</math>
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| which involves fractions. According to the theorem this is the only way to represent <math>P(x_1,x_2)</math> in terms of ''p''<sub>1</sub> and ''p''<sub>2</sub>. Therefore, ''P'' does not belong to the integral polynomial ring <math>\mathbb Z[p_1,\ldots,p_n].</math>
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| For another example, the [[elementary symmetric polynomial]]s ''e''<sub>''k''</sub>, expressed as polynomials in the power sum polynomials, do not all have integral coefficients. For instance,
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| :<math>e_2 := \sum_{1 \leq i<j \leq n} x_ix_j = \frac{p_1^2-p_2}{2} \, .</math>
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| The theorem is also untrue if the field has characteristic different from 0. For example, if the field ''F'' has characteristic 2, then <math>p_2 = p_1^2</math>, so ''p''<sub>1</sub> and ''p''<sub>2</sub> cannot generate ''e''<sub>2</sub> = ''x''<sub>1</sub>''x''<sub>2</sub>.
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| ''Sketch of a partial proof of the theorem'': By [[Newton's identities]] the power sums are functions of the elementary symmetric polynomials; this is implied by the following [[recurrence relation]], though the explicit function that gives the power sums in terms of the ''e''<sub>''j''</sub> is complicated (see [[Newton's identities]]):
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| :<math>p_n = \sum_{j=1}^n (-1)^{j-1} e_j p_{n-j} \,.</math>
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| Rewriting the same recurrence, one has the elementary symmetric polynomials in terms of the power sums (also implicitly, the explicit formula being complicated):
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| :<math> e_n = \frac{1}{n} \sum_{j=1}^n (-1)^{j-1} e_{n-j} p_j \,.</math>
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| This implies that the elementary polynomials are rational, though not integral, linear combinations of the power sum polynomials of degrees 1, ..., ''n''.
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| 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 <math>f(p_1,\ldots,p_n)</math> of the power sum symmetric polynomials ''p''<sub>1</sub>, ..., ''p''<sub>''n''</sub>. That is, the ring of symmetric polynomials is contained in the ring generated by the power sums, <math>\mathbb Q[p_1,\ldots,p_n].</math> Because every power sum polynomial is symmetric, the two rings are equal.
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| (This does not show how to prove the polynomial ''f'' is unique.)
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| For another system of symmetric polynomials with similar properties see [[complete homogeneous symmetric polynomial]]s.
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| ==References==
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| * Macdonald, I.G. (1979), ''Symmetric Functions and Hall Polynomials''. Oxford Mathematical Monographs. Oxford: Clarendon Press.
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| * Macdonald, I.G. (1995), ''Symmetric Functions and Hall Polynomials'', second ed. Oxford: Clarendon Press. ISBN 0-19-850450-0 (paperback, 1998).
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| * Richard P. Stanley (1999), ''Enumerative Combinatorics'', Vol. 2. Cambridge: Cambridge University Press. ISBN 0-521-56069-1
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| ==See also==
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| *[[Representation theory]]
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| *[[Newton's identities]]
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| [[Category:Homogeneous polynomials]]
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| [[Category:Symmetric functions]]
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