## Exhibit a generic symmetric polynomial

Exhibit the most general symmetric polynomial in the variables $x_1,x_2,x_3$ whose total degree is 4.

Recall that every symmetric polynomial in $x_1,x_2,x_3$ is a polynomial in the elementary symmetric polynomials $\sigma_1 = x_1+x_2+x_3$, $\sigma_2 = x_1x_2 + x_1x_3 + x_2x_3$, and $\sigma_3 = x_1x_2x_3$. (With $\sigma_0 = 1$, these are all of the elementary symmetric polynomials in three variables.) Note that the total degree of $\sigma_k$ is $k$; in fact, $\sigma_k$ is a $k$-form (all the terms of $\sigma_k$ have total degree $k$). Thus $\prod \sigma_{k_i}$ is a $\sum k_i$-form.

Any term in $F[\sigma_1,\sigma_2,\sigma_3]$ such that the sum of the indices on the “variables” $\sigma_i$ exceeds 4 results in a symmetric polynomial whose degree is larger than 4. The (monic) terms not violating this provision are as follows: $\sigma_3\sigma_1$, $\sigma_2\sigma_2$, $\sigma_2\sigma_1\sigma_1$, $\sigma_1\sigma_1\sigma_1\sigma_1$, $\sigma_3$, $\sigma_2\sigma_1$, $\sigma_1\sigma_1\sigma_1$, $\sigma_2$, $\sigma_1\sigma_1$, $\sigma_1$, and $\sigma_0$. Any linear combination of these terms is a symmetric polynomial of total degree 4 (provided one of the first 4 coefficients is nonzero). Moreover, every symmetic polynomial in $x_1$, $x_2$, and $x_3$ is of this form.