Editing Symmetric polynomial (section)

=== Monomial symmetric polynomials === <!--[[Monomial symmetric polynomial]] redirects here -->
Powers and products of elementary symmetric polynomials work out to rather complicated expressions. If one seeks basic ''additive'' building blocks for symmetric polynomials, a more natural choice is to take those symmetric polynomials that contain only one type of [[monomial]], with only those copies required to obtain symmetry. Any monomial in ''X''<sub>1</sub>, ..., ''X''<sub>''n''</sub> can be written as ''X''<sub>1</sub><sup>α<sub>1</sub></sup>...''X''<sub>''n''</sub><sup>α<sub>''n''</sub></sup> where the exponents α<sub>''i''</sub> are [[natural number]]s (possibly zero); writing α&nbsp;=&nbsp;(α<sub>1</sub>,...,α<sub>''n''</sub>) this can be abbreviated to ''X''<sup>&thinsp;α</sup>. The '''monomial symmetric polynomial''' ''m''<sub>α</sub>(''X''<sub>1</sub>, ..., ''X''<sub>''n''</sub>) is defined as the sum of all monomials ''x''<sup>β</sup> where β ranges over all ''distinct'' permutations of (α<sub>1</sub>,...,α<sub>''n''</sub>). For instance one has
:<math>m_{(3,1,1)}(X_1,X_2,X_3)=X_1^3X_2X_3+X_1X_2^3X_3+X_1X_2X_3^3</math>,
:<math>m_{(3,2,1)}(X_1,X_2,X_3)=X_1^3X_2^2X_3+X_1^3X_2X_3^2+X_1^2X_2^3X_3+X_1^2X_2X_3^3+X_1X_2^3X_3^2+X_1X_2^2X_3^3.</math>
Clearly ''m''<sub>α</sub>&nbsp;=&nbsp;''m''<sub>β</sub> when β is a permutation of α, so one usually considers only those ''m''<sub>α</sub> for which α<sub>1</sub>&nbsp;≥&nbsp;α<sub>2</sub>&nbsp;≥&nbsp;...&nbsp;≥&nbsp;α<sub>''n''</sub>, in other words for which α is a [[partition (number theory)|partition of an integer]].
These monomial symmetric polynomials form a vector space [[basis (linear algebra)|basis]]: every symmetric polynomial ''P'' can be written as a [[linear combination]] of the monomial symmetric polynomials. To do this it suffices to separate the different types of monomial occurring in ''P''. In particular if ''P'' has integer coefficients, then so will the linear combination.

The elementary symmetric polynomials are particular cases of monomial symmetric polynomials: for 0&nbsp;≤&nbsp;''k''&nbsp;≤&nbsp;''n'' one has
:<math>e_k(X_1,\ldots,X_n)=m_\alpha(X_1,\ldots,X_n)</math> where &alpha; is the partition of ''k'' into ''k'' parts&nbsp;1 (followed by ''n''&nbsp;−&nbsp;''k'' zeros).