Polynomial Discriminant

A polynomial discriminant is the product of the squares of the differences of the polynomial roots . The discriminant of a polynomial is defined only up to constant factor, and several slightly different normalizations can be used. For a polynomial

(1)

of degree , the most common definition of the discriminant is

(2)

which gives a homogenous polynomial of degree in the coefficients of .

The discriminant of a polynomial is given in terms of a resultant as

(3)

where is the derivative of and is the degree of . For fields of infinite characteristic, so the formula reduces to

(4)

The discriminant of a univariate polynomial is implemented in the Wolfram Language as Discriminant[p, x].

The discriminant of the quadratic equation

(5)

is given by

(6)

The discriminant of the cubic equation

(7)

is given by

(8)

The discriminant of a quartic equation

(9)

is

(10)

(Schroeppel 1972).