The best reference I can find is Factorization of Cyclic and Symmetric polynomials. I want to get the whole picture of the basic elements of the construction of the polynomials, so I can apply them to dimensional analysis.

As an example, the derivation of Heron's formula: A triangle had sides lengths $a$, $b$, and $c$. Denote the area by $S$. First we let $b=c=1$, then the height on base $a$ is $ \sqrt{1-\frac{a^2}{4}} $. Therefore, $S= \frac12 a \sqrt{1-\frac{a^2}{4}} $, i.e., $ S^2= \frac{1}{16}(-a^4+4a^2) \tag{1} $

Now consider the general case. Since $a$ is an independent variable, the formula for $S^2$ should be 4 quartic ternary polynomials of $a$, $b$, and $c$. Because $a$, $b$ and $c$ are cyclic, if there is an $a^3$ term in the formula for $S^2$ then there must be an $a^3(b+c)$ term. Since $a^3$ will not vanish when $b=c$, from (1) we know there are no $a^3$ terms in formula of $S^2$, likewise for terms of $a$. We can therefore set $$S^2=m(a^4+b^4+c^4)+n(a^2b^2+b^2c^2+c^2a^2) $$ and solve for the coefficients $m=-\frac{1}{16}$, $n=\frac{1}{8} $.

My question is, what are the building blocks in $n$-degree symmetric, anti-symmetric, and cyclic polynomials? I am sure there are general discussion of this topics but I had difficult to find them.



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