Any symmetric polynomial (respectively, symmetric rational function) can be expressed as a polynomial (respectively, rational
function) in the elementary symmetric polynomials
on those variables.
There is a generalization of this theorem to polynomial invariants of permutation groups, which states that any polynomial invariant can be represented
as a finite linear combination of special -invariant orbit polynomials with symmetric functions as coefficients,
i.e.,
where ,
and , ..., are elementary symmetric functions, and , ..., are special terms. Furthermore, any special term
has a total degree , and a maximal variable degree .
Coolidge, J. L. A Treatise on Algebraic Plane Curves. New York: Dover, p. 2, 1959.Göbel,
M. "Computing Bases for Permutation-Invariant Polynomials." J. Symb.
Comput.19, 285-291, 1995.Göbel, M. "On the Number
of Special Permutation-Invariant Orbits and Terms." Appl. Algebra Eng. Comm.
Comput.8, 505-509, 1997.Herstein, I. N. Noncommutative
Rings. Washington, DC: Math. Assoc. Amer., 1968.
, which states that any polynomial invariant ![f in R[X_1,...,X_n]](/images/equations/FundamentalTheoremofSymmetricFunctions/Inline2.svg)
can be represented
as a finite linear combination of special 
-invariant orbit polynomials with symmetric functions as coefficients,
i.e.,
![p_t in R[X_1,...,X_n]](/images/equations/FundamentalTheoremofSymmetricFunctions/Inline4.svg)
,
, ..., 
are elementary symmetric functions, and 
, ..., 
are special terms. Furthermore, any special term 
has a total degree 
, and a maximal variable degree 
.
