Writing this difference inductively as a polynomial in the elementary symmetric polynomials, and adding back to it, one obtains the sought for polynomial expression for.
12.
There is one elementary symmetric polynomial of degree in variables for each nonnegative integer, and it is formed by adding together all distinct products of distinct variables.
13.
An example of such relations are Newton's identities, which express the sum of any fixed power of the roots in terms of the elementary symmetric polynomials.
14.
The fact that this expression is unique, or equivalently that all the products ( monomials ) of elementary symmetric polynomials are linearly independent, is also easily proved.
15.
And so on, that allows to recursively express the successive complete homogeneous symmetric polynomials in terms of the elementary symmetric polynomials; the latter gives a set of equations
16.
Now one proves by induction on the leading monomial in lexicographic order, that any nonzero homogeneous symmetric polynomial of degree can be written as polynomial in the elementary symmetric polynomials.
17.
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.
18.
*PM : elementary symmetric polynomial in terms of power sums, id = 7965 new !-- WP guess : elementary symmetric polynomial in terms of power sums-- Status:
19.
*PM : elementary symmetric polynomial in terms of power sums, id = 7965 new !-- WP guess : elementary symmetric polynomial in terms of power sums-- Status:
20.
More specifically, a Hessian equation is the " k "-trace, or the " k " th elementary symmetric polynomial of eigenvalues of the Hessian matrix.
