Is there a function of the coefficients of a polynomial (probably not a polynomial in the coefficients) that can determine the number of complex root-pairs mod 3? I'm after something roughly analogous to what the discriminant does for the number of root-pairs mod 2 or an understanding of why it can't exist.

The determinant of real polynomial of degree $n$ with leading coefficient $k$ is given by:

$$ k^\left(2n-2\right)\;\prod_{1 \le i \le n}\; \prod_{1 \le j \le n} (r_i-r_j)^{(2\cdot[i < j])} $$

Since non-real roots come in conjugate pairs, it seems clear that the discriminant determines the parity of the count of non-real pairs. (Hence the ambiguity in the quartic case between all real solutions and no real solutions both map to a positive number).

Furthermore, since the discriminant is a symmetric polynomial of the roots, then it is a polynomial over the elementary symmetric polynomials of the roots.


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