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.
Read more about this topic: Ring Of Symmetric Functions
Famous quotes containing the words ring and/or functions:
“When I received this [coronation] ring I solemnly bound myself in marriage to the realm; and it will be quite sufficient for the memorial of my name and for my glory, if, when I die, an inscription be engraved on a marble tomb, saying, “Here lieth Elizabeth, which reigned a virgin, and died a virgin.””
—Elizabeth I (1533–1603)
“The mind is a finer body, and resumes its functions of feeding, digesting, absorbing, excluding, and generating, in a new and ethereal element. Here, in the brain, is all the process of alimentation repeated, in the acquiring, comparing, digesting, and assimilating of experience. Here again is the mystery of generation repeated.”
—Ralph Waldo Emerson (1803–1882)