Sylvester's Discriminant

In 1851, Sylvester discovered a criterion for cubic equations that allows statements to be made about the number and type of solutions. The characterisation was done by an expression, for which he coined the term discriminant.

The criterion for quadratic equations is as follows:

A quadratic equation ax2+bx+c=0ax^2+bx+c=0 has two real solutions, if for the coefficients a,b,ca, b, c the discriminant Δ=b2−4ac>0\Delta = b^2-4ac >0. It only has a single real solution if Δ=0\Delta =0 and no real solution if Δ<0\Delta <0. Here the discriminant Δ\Delta consists of two summands, each with degree 2.

Properties of solutions of cubic equations ax3+bx2+cx+d=0ax^3+bx^2+cx+d=0 can be determined by the discriminant Δ=b2c2−4ac3−4b3d−27a2d2+18abcd\Delta =b^2c^2-4ac^3-4b^3d-27a^2d^2+18abcd a term whose five summands each have degree 4.

If Δ>0\Delta >0 then the equation has three distinct real solutions; if Δ<0\Delta <0 then the equation has one real and two conjugate complex solutions; if Δ=0\Delta =0 then all solutions are real, but at least two of them match.

The discriminants for higher degree equations consist of an exponentially growing numbers of summands (4th degree: 16; 5th degree: 59; 6th degree: 246).

The terms can be obtained by calculating the determinants of the corresponding Sylvester matrices:

The first n−1n-1 lines of these matrices consist of the coefficients of the polynomial and the next nn lines come from the coefficients of the 1st derivation of the polynomial:

Second degree:
(−1)⋅a⋅Δ2=∣abc2ab002ab∣=ab2+4a2c−2ab2(-1) \cdot a\cdot \Delta _2 = \begin {vmatrix} a & b & c \\ 2a & b & 0 \\ 0 & 2a & b \\ \end {vmatrix} =ab^2+4a^2c-2ab^2
=4a2c−ab2=a⋅(4ac−b2)=4a^2c-ab^2 =a \cdot (4ac-b^2)

Third degree:
(−1)3⋅a⋅Δ3=∣abcd00abcd3a2bc0003a2bc0003a2bc∣=...(-1)^3 \cdot a\cdot \Delta _3 = \begin {vmatrix} a & b & c & d & 0 \\ 0 & a & b & c & d \\ 3a & 2b & c & 0 & 0 \\ 0 & 3a & 2b & c & 0 \\ 0 & 0 & 3a & 2b & c \\ \end {vmatrix} = ...=a⋅(b2c2−4ac3−4b3d−27a2d2+18abcd)=a \cdot (b^2c^2-4ac^3-4b^3d-27a^2d^2+18abcd)

The expressions become simpler if the polynomials are normalised (i.e. a=1a=1) or changed by a substitution so that the summand of the second highest power vanishes.

Thanks to H K Strick