Discriminants of symmetric polynomials

A homogeneous polynomial $$S(x_1, \ldots, x_n)$$ of degree $$r$$ in $$n$$ variables possesses a discriminant $$D_{n|r}(S)$$ , which vanishes if and only if the system of equations $$\partial S / \partial x_i = 0$$ has nontrivial solutions. We provide an explicit formula for the discriminants of symmetric (under permutations of $$x_1, \ldots, x_n$$ ) homogeneous polynomials of degree $$r$$ in $$n \geq r$$ variables. This formula is highly effective from a computational perspective: symbolic computer calculations using this formula take seconds even for $$n \approx 20$$ . We work out the cases $$r = 2$$ , $$3$$ , and $$4$$ in detail. We also consider the case of completely antisymmetric polynomials.