  A homogeneous polynomial S(x_1, ..., x_n) of degree r in n variables posesses
a discriminant D_{n|r}(S), which vanishes if and only if the system of
equations dS/dx_i = 0 has non-trivial solutions. We give an explicit formula
for discriminants of symmetric (under permutations of x_1, ..., x_n)
homogeneous polynomials of degree r in n >= r variables. This formula is
division free and quite effective from the computational point of view:
symbolic computer calculations with the help of this formula take seconds even
for n ~ 20. We work out in detail the cases r = 2,3,4 which will be probably
important in applications. We also consider the case of completely
antisymmetric polynomials.

Algebraic Geometry