reduction algorithm for symmetric polynomials

We give here an algorithm for reducing a symmetric polynomialMathworldPlanetmath into a polynomialMathworldPlanetmathPlanetmathPlanetmath in the elementary symmetric polynomials.

We define the height of a monomialMathworldPlanetmathPlanetmath x1e1xnen in R[x1,,xn] to be e1+2e2++nen. The height of a polynomial is defined to be the maximum height of any of its monomial terms, or 0 if it is the zero polynomialMathworldPlanetmath.

Let f be a symmetric polynomial. We reduce f into elementary symmetric polynomials by inductionMathworldPlanetmath on the height of f. Let cx1e1xnen be the monomial term of maximal height in f. Consider the polynomial


where sk is the k–th elementary symmetric polynomial in the n variables x1,,xn. Then g is a symmetric polynomial of lower height than f, so by the induction hypothesis, g is a polynomial in s1,,sn, and it follows immediately that f is also a polynomial in s1,,sn.

Title reduction algorithm for symmetric polynomials
Canonical name ReductionAlgorithmForSymmetricPolynomials
Date of creation 2013-03-22 12:11:17
Last modified on 2013-03-22 12:11:17
Owner djao (24)
Last modified by djao (24)
Numerical id 6
Author djao (24)
Entry type Proof
Classification msc 05E05
Classification msc 12F10
Defines height