action of (every element of) ${\mathcal{S}}_n$. It is well known that the ring $T=A[x_1,\cdots,x_n]^{{\mathcal{S}}_n}$ of symmetric polynomials is given by $$\begin{aligned}
\label{symsym}
T=A[x_1,\cdots,x_n]^{{\mathcal{S}}_n} &=&A[e_1,\cdots,e_n],\end{aligned}$$ where $e_1, \cdots, e_n$ are the so-called [*elementary symmetric polynomials*]{} defined by $$e_j = \sum_{1 \le i_1 < i_2 < \cdots < i_j \le n}  x_{i_1} x_{i_2} \cdots x_{i_j};$$ see [@Lang Theorem 6.1, p. 191].

A non-zero polynomial $\Phi$ in $B$ is said to be [*homogeneous*]{} if all of its terms are of the same (total) degree $d$. More precisely, if $$\Phi (\lambda x_1, \cdots, \lambda x_n) = \lambda^d \Phi  
(x_1, \cdots, x_n)$$ for any variable $\lambda$.

It is clear that every non-zero polynomial $\Phi  \in B$ of (total) degree $n \ge 0$ can be written uniquely in the form $$\Phi  = h^{(0)} + h^{(1)} + \cdots + h^{(n)},$$ where $h^{(j)}$ is either 0 or homogeneous of degree $j$, and $h^{(n)}$ is not zero. The last term $h^{(n)}$ is called the [*leading homogeneous component*]{} of $h$, and will be denoted by $\Phi^*$. Notice that $$\begin{aligned}
\deg \Phi^* = \deg \Phi.
\label{degh}\end{aligned}$$

If $A$ is an integral domain, then it is easy to see that $$\begin{aligned}
(\Phi \Psi)^* = \Phi^* \Psi^*.
\label{fgh}\end{aligned}$$ Then it follows from (\[degh\]) 