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.

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