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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).
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.
But the terms of P which contain only the variables X 1, ..., X n − 1 are precisely the terms that survive the operation of setting X n to 0, so their sum equals P(X 1, ..., X n − 1, 0), which is a symmetric polynomial in the variables X 1, ..., X n − 1 that we shall denote by P̃(X 1, ..., X n − 1). By the inductive hypothesis, this ...
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.
Multiplying this by the generating function for the complete homogeneous symmetric polynomials, one obtains the constant series 1 (equivalently, plethystic exponentials satisfy the usual properties of an exponential), and the relation between the elementary and complete homogeneous polynomials follows from comparing coefficients of t m.
(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 ...
In the mathematics of operator theory, an operator A on an (infinite dimensional) Banach space or Hilbert space H has a cyclic vector f if the vectors f, Af, A 2 f,... span H. Equivalently, f is a cyclic vector for A in case the set of all vectors of the form p(A)f, where p varies over all polynomials, is dense in H. [1] [2]
An nth root of unity is a complex number whose nth power is 1, a root of the polynomial x n − 1. The set of all nth roots of unity forms a cyclic group of order n under multiplication. [1] The generators of this cyclic group are the nth primitive roots of unity; they are the roots of the nth cyclotomic polynomial.