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Symmetric polynomials

Elementary Symmetrical Polynomials

The elementary symmetrical polynomials (ESP's) are polynomials in n indeterminates that are symmetrical, that is, they don't change under the permuation of the indeterminates.

They are defined as follows.

S1(t1, t2, $\cdots$, tn) = t1 + t2 + $\cdots$ + tn
S2(t1, t2, $\cdots$, tn) = t1t2 + t1t3 + $\cdots$ + t1tn + t2t3 + t2t4 + $\cdots$ + t2tn + t3t4 + $\cdots$ + tn-1tn
S3(t1, t2, $\cdots$, tn) = t1t2t3 + t1t2t4 + $\cdots$ + t1t2tn + $\cdots$ + t1tn-1tn + $\cdots$ + tn-2tn-1tn
Sn(t1, t2, $\cdots$, tn) = t1t2 $\cdots$tn

In other words, Sk exists of a sum of all permutations of products of precisely k different tj. There are $\binom{n}{k} = \frac{n!}{(n-k)!k!} $ such permutations for a given n and k.

A more formal way of defining the ESP's is

\[ S_k(t_1, t_2, \ldots, t_n) = \sum_{1 \leq i_1 < i_2 < \cdots < i_k \leq n}{t_{i_1}t_{i_2} \cdots t_{i_k}} \]

Newton Symmetrical Polynomials

The Newton Symmetric Polynomials (NSP's) on (t1 t2 $\cdots$ tn) are the polynomials

\[ N_k(t_1, t_2, \ldots, t_n) = \sum_{i=1}^n{t_i^k} \]

Newton and Elementary Symmetrical Polynomials

For a fixed n, the set { S1, S2, ..., Sn } is linearly independend; they cannot be expressed into eachother. Since there are n indeterminates and n elementary symmetric polynomials, they form a basis for the set of symmetric polynomials of n indeterminates. The same holds for the set { N1, N2, ..., Nn }. Every symmetric polynomial can be expressed as a polynomial of the ESP's or NSP's, in particular they can be expressed into eachother.

A recursive formula that expresses one into the other still looks reasonably simple. Therefore this is the prefered way. For a fixed n the relationship is given by

\[ S_k = \frac{1}{k}(S_{k-1}N_1 - S_{k-2}N_2 + \ldots \pm S_1N_{k-1} \mp N_k) \]

from which it is easy to derive the reverse relationship

\[ N_k = S_1N_{k-1} - S_2N_{k-2} + \ldots \pm S_{k-1}N_1 \mp kS_k \]

Both cases exist of a sum with k - 1 leading terms, followed by the last term. In particular, N1 = S1.

Copyright © 2002-2008 Carlo Wood.  All rights reserved.