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[parent] reduction algorithm for symmetric polynomials (Proof)

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

We define the height of a monomial $ x_1^{e_1} \cdots x_n^{e_n}$ in $ R[x_1,\dots,x_n]$ to be $ e_1 + 2 e_2 + \cdots + n e_n$. 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 polynomial.

Let $ f$ be a symmetric polynomial. We reduce $ f$ into elementary symmetric polynomials by induction on the height of $ f$. Let $ c x_1^{e_1} \cdots x_n^{e_n}$ be the monomial term of maximal height in $ f$. Consider the polynomial

$\displaystyle g := f - c s_1^{e_n - e_{n-1}} s_2^{e_{n-1} - e_{n-2}} \cdots s_{n-1}^{e_2 - e_1} s_n^{e_1} $
where $ s_k$ is the $ k$-th elementary symmetric polynomial in the $ n$ variables $ x_1, \dots, x_n$. Then $ g$ is a symmetric polynomial of lower height than $ f$, so by the induction hypothesis, $ g$ is a polynomial in $ s_1, \dots, s_n$, and it follows immediately that $ f$ is also a polynomial in $ s_1, \dots, s_n$.



"reduction algorithm for symmetric polynomials" is owned by djao.
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Cross-references: induction hypothesis, variables, induction, zero polynomial, terms, height of a polynomial, monomial, elementary symmetric polynomials, polynomial, symmetric polynomial, algorithm
There are 5 references to this entry.

This is version 2 of reduction algorithm for symmetric polynomials, born on 2002-01-08, modified 2003-06-28.
Object id is 1454, canonical name is ReductionAlgorithmForSymmetricPolynomials.
Accessed 5463 times total.

Classification:
AMS MSC: 05E05 (Combinatorics :: Algebraic combinatorics :: Symmetric functions)
 12F10 (Field theory and polynomials :: Field extensions :: Separable extensions, Galois theory)
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