Why does it mean that $n$-th variable is removable?

Question

I'm reading the proof for "the fundamental theorem of symmetric polynomials" and I have a trouble with it (http://en.m.wikipedia.org/wiki/Elementary_symmetric_polynomial)

Let $P(X_1,...,X_n)$ be a symmetric homogeneous polynomials in variables $X_1,...,X_n$.

Then, $P(X_1,...,X_n)$ can be decomposed as a sum of symmetric homogeneous polynomials, that is, $P(X_1,...,X_n)=P_{\text{lacunary}}(X_1,...,X_n)+ X_1...X_nQ(X_1,...,X_n)$, where lacunary part is defined as the sum of all monomials in $P$ which contain only a proper subset of variables $X_1,...,X_n$.

Then, it's written in wikipedia that "since the lacunary part is symmetric, $P_{\text{lacunary}}$ is determined by $X_1,...,X_{n-1}$."

How?

Let $\Phi$ be the endomorphism on $A[X_1,...,X_n]$ such that $\Phi$ fixes $X_i$ for $1\leq i\leq n-1$ and $\Phi(X_n)=0$.

Then, $P(X_1,...,X_{n-1},0)=\Phi(P(X_1,...,X_n))=\Phi(P_{\text{lacunary}}(X_1,...,X_n))+ \Phi(X_1...X_nQ(X_1,...,X_n))= \Phi(P_{\text{lacunary}}(X_1,...,X_n))$

However, I don't get how $\Phi(P_{\text{lacunary}}(X_1,...,X_n))$ is equal to $P_{\text{lacunary}}(X_1,...,X_n)$. How do I assert it?

Thank you in advance

Answer 1

Let's decompose $P$ into distinct monomials, say, $P=a_1+ ... + a_k + b_1 + ... + b_l$ where $a_1+ ... + a_k$ is the lacunary part of $P$.

Let $\Phi_\sigma$ be the automorphism on $R[X_1,...,X_n]$ corresponding to a permutation $\sigma\in S_n$.

Then, $\Phi_\sigma( a_1 ) + ... + \Phi_\sigma( a_k ) + \Phi_\sigma( b_1 ) + ... \Phi_\sigma( b_l ) = \Phi_\sigma(P)=P=a_1 + ... + a_k + b_1 + ... + b_l$.

This means that $\Phi_\sigma$ permutes the monomials of $P$ and this implies that $\Phi_\sigma$ fixes lacunary part.

Now note that $P(X_1,...,X_{n-1},0)$ is the sum of terms in the lacunary part which lack in $X_n$ and there is a one-to-one correspondence between $\{ \text{ terms of the lacunary part lacking in } X_i \}$ and $\{ \text{ terms of the lacunary part lacking in } X_n \}$. Hence by knowing if sums of terms lacking in $X_n$ of two different homogeneous polynomials are equal, then the whole lacunary parts of two are equal.