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The following polynomials in two variables X 1 and X 2 are symmetric: + + + + (+) as is the following polynomial in three variables X 1, X 2, X 3: . There are many ways to make specific symmetric polynomials in any number of variables (see the various types below).
That is, any symmetric polynomial P is given by an expression involving only additions and multiplication of constants and elementary symmetric polynomials. There is one elementary symmetric polynomial of degree d in n variables for each positive integer d ≤ n, and it is formed by adding together all distinct products of d distinct variables.
The complete homogeneous symmetric polynomials are characterized by the following identity of formal power series in t: = (, …,) = = = = = (this is called the generating function, or generating series, for the complete homogeneous symmetric polynomials).
Symmetric polynomials arise naturally in the study of the relation between the roots of a polynomial in one variable and its coefficients, since the coefficients can be given by polynomial expressions in the roots, and all roots play a similar role in this setting. From this point of view, the elementary symmetric polynomials are the most ...
The Newton identities also permit expressing the elementary symmetric polynomials in terms of the power sum symmetric polynomials, showing that any symmetric polynomial can also be expressed in the power sums. In fact the first n power sums also form an algebraic basis for the space of symmetric polynomials.
(here Λ n denotes the ring of symmetric polynomials in n indeterminates), and also in (Stanley, 1999). To define a symmetric function one must either indicate directly a power series as in the first construction, or give a symmetric polynomial in n indeterminates for every natural number n in a way compatible with the second construction. An ...
A basic fact on symmetric polynomials is that any symmetric polynomial in, say, t i 's is a polynomial in elementary symmetric polynomials in t i 's. Either by splitting principle or by ring theory, any Chern polynomial c t ( E ) {\displaystyle c_{t}(E)} factorizes into linear factors after enlarging the cohomology ring; E need not be a direct ...
Aside from polynomial functions, tensors that act as functions of several vectors can be symmetric, and in fact the space of symmetric -tensors on a vector space is isomorphic to the space of homogeneous polynomials of degree on . Symmetric functions should not be confused with even and odd functions, which have a different sort of symmetry.