Elementary Symmetric Polynomial - The Fundamental Theorem of Symmetric Polynomials - Proof Sketch

Proof Sketch

The theorem may be proved for symmetric homogeneous polynomials by a double mathematical induction with respect to the number of variables n and, for fixed n, with respect to the degree of the homogeneous polynomial. The general case then follows by splitting an arbitrary symmetric polynomial into its homogeneous components (which are again symmetric).

In the case n = 1 the result is obvious because every polynomial in one variable is automatically symmetric.

Assume now that the theorem has been proved for all polynomials for variables and all symmetric polynomials in n variables with degree < d. Every homogeneous symmetric polynomial P in can be decomposed as a sum of homogeneous symmetric polynomials

Here the "lacunary part" is defined as the sum of all monomials in P which contain only a proper subset of the n variables X₁, ..., X_n, i.e., where at least one variable X_j is missing.

Because P is symmetric, the lacunary part is determined by its terms containing only the variables X₁, ..., X_n−1, i.e., which do not contain X_n. These are precisely the terms that survive the operation of setting X_n to 0, so their sum equals, which is a symmetric polynomial in the variables X₁, ..., X_n−1 that we shall denote by . By the inductive assumption, this polynomial can be written as

for some . Here the doubly indexed denote the elementary symmetric polynomials in n−1 variables.

Consider now the polynomial

Then is a symmetric polynomial in X₁, ..., X_n, of the same degree as, which satisfies

(the first equality holds because setting X_n to 0 in gives, for all ), in other words, the lacunary part of R coincides with that of the original polynomial P. Therefore the difference P−R has no lacunary part, and is therefore divisible by the product of all variables, which equals the elementary symmetric polynomial . Then writing, the quotient Q is a homogeneous symmetric polynomial of degree less than d (in fact degree at most d − n) which by the inductive assumption can be expressed as a polynomial in the elementary symmetric functions. Combining the representations for P−R and R one finds a polynomial representation for P.

The uniqueness of the representation can be proved inductively in a similar way. (It is equivalent to the fact that the n polynomials are algebraically independent over the ring A.) The fact that the polynomial representation is unique implies that is isomorphic to .