SymmetricPolynomial

SymmetricPolynomial[k,{x1,,xn}]

gives the k^(th) elementary symmetric polynomial in the variables x1,,xn.

Details

  • A symmetric polynomial of n variables {x1,,xn} is invariant under any permutation of its variables. The k^(th) elementary symmetric polynomial is the sum of all square-free monomials of degree k.
  • The degree k must satisfy 0kn.
  • The elementary symmetric polynomials form a basis for the symmetric polynomials.
  • Expressing a general symmetric polynomial in terms of elementary symmetric polynomials is accomplished by using SymmetricReduction.

Examples

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Basic Examples  (1)

The elementary symmetric polynomial of degree 3 in variables x1,x2,x3,x4:

Scope  (1)

The zeroth elementary symmetric polynomial is defined to be 1:

Applications  (1)

The 2×3 matrices with entries 0 or 1:

Select matrices whose column sums are 1,1,1 and whose row sums are 2,1:

You can also count how many such matrices there are by using SymmetricPolynomial. The generating function of 2×3 matrices whose row sums are 2,1 is given by:

The coefficient of x11x21x31 counts how many of these matrices have column sums 1,1,1:

Properties & Relations  (4)

The generating function for the symmetric polynomials in variables is given by :

Check:

The monic polynomial with roots has coefficients that are elementary symmetric polynomials of the :

The elementary symmetric polynomials ek=SymmetricPolynomial[k,{x1,,xn}] are related to the power sum polynomials through the NewtonGirard identities:

For example, with :

Neat Examples  (1)

Find integers such that the roots of are :

Check:

Introduced in 2007
 (6.0)