The Ring of Symmetric Functions
Most relations between symmetric polynomials do not depend on the number n of indeterminates, other than that some polynomials in the relation might require n to be large enough in order to be defined. For instance the Newton's identity for the third power sum polynomial leads to
where the denote elementary symmetric polynomials; this formula is valid for all natural numbers n, and the only notable dependency on it is that ek(X1,…,Xn) = 0 whenever n < k. One would like to write this as an identity p3 = e13 − 3e2e1 + 3e3 that does not depend on n at all, and this can be done in the ring of symmetric polynomials. In that ring there are elements ek for all integers k ≥ 1, and an arbitrary element can be given by a polynomial expression in them.
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