Editing Symmetric polynomial (section)

=== Complete homogeneous symmetric polynomials ===
{{Main|Complete homogeneous symmetric polynomial}}
For each nonnegative integer ''k'', the complete homogeneous symmetric polynomial ''h''<sub>''k''</sub>(''X''<sub>1</sub>, ..., ''X''<sub>''n''</sub>) is the sum of all distinct [[monomial]]s of [[degree of a polynomial|degree]] ''k'' in the variables ''X''<sub>1</sub>, ..., ''X''<sub>''n''</sub>. For instance
:<math>h_3(X_1,X_2,X_3) = X_1^3+X_1^2X_2+X_1^2X_3+X_1X_2^2+X_1X_2X_3+X_1X_3^2+X_2^3+X_2^2X_3+X_2X_3^2+X_3^3.</math>
The polynomial ''h''<sub>''k''</sub>(''X''<sub>1</sub>, ..., ''X''<sub>''n''</sub>) is also the sum of all distinct monomial symmetric polynomials of degree ''k'' in ''X''<sub>1</sub>, ..., ''X''<sub>''n''</sub>, for instance for the given example
:<math>\begin{align}
 h_3(X_1,X_2,X_3)&=m_{(3)}(X_1,X_2,X_3)+m_{(2,1)}(X_1,X_2,X_3)+m_{(1,1,1)}(X_1,X_2,X_3)\\
 &=(X_1^3+X_2^3+X_3^3)+(X_1^2X_2+X_1^2X_3+X_1X_2^2+X_1X_3^2+X_2^2X_3+X_2X_3^2)+(X_1X_2X_3).\\
\end{align}</math>

All symmetric polynomials in these variables can be built up from complete homogeneous ones: any symmetric polynomial in ''X''<sub>1</sub>, ..., ''X''<sub>''n''</sub> can be obtained from the complete homogeneous symmetric polynomials ''h''<sub>1</sub>(''X''<sub>1</sub>, ..., ''X''<sub>''n''</sub>), ..., ''h''<sub>''n''</sub>(''X''<sub>1</sub>, ..., ''X''<sub>''n''</sub>) via multiplications and additions. More precisely:
:Any symmetric polynomial ''P'' in ''X''<sub>1</sub>, ..., ''X''<sub>''n''</sub> can be written as a polynomial expression in the polynomials ''h''<sub>''k''</sub>(''X''<sub>1</sub>, ..., ''X''<sub>''n''</sub>) with 1&nbsp;&le;&nbsp;''k''&nbsp;&le;&nbsp;''n''.
:If ''P'' has integral coefficients, then the polynomial expression also has integral coefficients.
For example, for ''n'' = 2, the relevant complete homogeneous symmetric polynomials are {{math|1=''h''<sub>1</sub>(''X''<sub>1</sub>, ''X''<sub>2</sub>) = ''X''<sub>1</sub> + ''X''<sub>2</sub>}} and {{math|1=''h''<sub>2</sub>(''X''<sub>1</sub>, ''X''<sub>2</sub>) = ''X''<sub>1</sub><sup>2</sup> + ''X''<sub>1</sub>''X''<sub>2</sub> + ''X''<sub>2</sub><sup>2</sup>}}. The first polynomial in the list of examples above can then be written as
:<math>X_1^3+ X_2^3-7 = -2h_1(X_1,X_2)^3+3h_1(X_1,X_2)h_2(X_1,X_2)-7.</math>
As in the case of power sums, the given statement applies in particular to the complete homogeneous symmetric polynomials beyond ''h''<sub>''n''</sub>(''X''<sub>1</sub>, ..., ''X''<sub>''n''</sub>), allowing them to be expressed in terms of the ones up to that point; again the resulting identities become invalid when the number of variables is increased.

An important aspect of complete homogeneous symmetric polynomials is their relation to elementary symmetric polynomials, which can be expressed as the identities
:<math>\sum_{i=0}^k(-1)^i e_i(X_1,\ldots,X_n)h_{k-i}(X_1,\ldots,X_n) = 0</math>, for all ''k''&nbsp;&gt;&nbsp;0, and any number of variables&nbsp;''n''.
Since ''e''<sub>0</sub>(''X''<sub>1</sub>, ..., ''X''<sub>''n''</sub>) and ''h''<sub>0</sub>(''X''<sub>1</sub>, ..., ''X''<sub>''n''</sub>) are both equal to&nbsp;1, one can isolate either the first or the last term of these summations; the former gives a set of equations that allows one to recursively express the successive complete homogeneous symmetric polynomials in terms of the elementary symmetric polynomials, and the latter gives a set of equations that allows doing the inverse. This implicitly shows that any symmetric polynomial can be expressed in terms of the ''h''<sub>''k''</sub>(''X''<sub>1</sub>, ..., ''X''<sub>''n''</sub>) with 1&nbsp;≤&nbsp;''k''&nbsp;≤&nbsp;''n'': one first expresses the symmetric polynomial in terms of the elementary symmetric polynomials, and then expresses those in terms of the mentioned complete homogeneous ones.