Discriminants of Symmetric Polynomials

TL;DR

This paper provides an explicit, computationally efficient formula for the discriminants of symmetric homogeneous polynomials, facilitating quick calculations for polynomials with up to 20 variables, and explores antisymmetric cases.

Contribution

It introduces a division-free formula for discriminants of symmetric polynomials, enabling fast symbolic computations and detailed analysis for degrees 2, 3, and 4.

Findings

Efficient formula for symmetric polynomial discriminants

Fast symbolic calculations for up to 20 variables

Detailed cases for degrees 2, 3, and 4

Abstract

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.