Exhibit the most general symmetric polynomial in the variables whose total degree is 4.
Recall that every symmetric polynomial in is a polynomial in the elementary symmetric polynomials , , and . (With , these are all of the elementary symmetric polynomials in three variables.) Note that the total degree of is ; in fact, is a -form (all the terms of have total degree ). Thus is a -form.
Any term in such that the sum of the indices on the “variables” exceeds 4 results in a symmetric polynomial whose degree is larger than 4. The (monic) terms not violating this provision are as follows: , , , , , , , , , , and . 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 , , and is of this form.