Discriminant

The discriminant Δ of an [Graphics:../Images/tricks_gr_362.gif]-th degree polynomial with roots [Graphics:../Images/tricks_gr_363.gif] is the symmetric product [Graphics:../Images/tricks_gr_364.gif]. (See "Algebra of polynomial roots," TMJ 6(2): 18.) One way of computing Δ is to obtain the explicit equations for the coefficients of the polynomial by comparing the expanded and factored forms using [Graphics:../Images/tricks_gr_365.gif] (implicitly via an order term [Graphics:../Images/tricks_gr_366.gif]) and [Graphics:../Images/tricks_gr_367.gif]. Consider the case [Graphics:../Images/tricks_gr_368.gif].

[Graphics:../Images/tricks_gr_369.gif]
[Graphics:../Images/tricks_gr_370.gif]

With the Groebner basis for the coefficients,

[Graphics:../Images/tricks_gr_371.gif]
[Graphics:../Images/tricks_gr_372.gif]

we can use [Graphics:../Images/tricks_gr_373.gif] to express the discriminant [Graphics:../Images/tricks_gr_374.gif] in terms of the coefficients β and γ.

[Graphics:../Images/tricks_gr_375.gif]
[Graphics:../Images/tricks_gr_376.gif]

The last term of this expression is the discriminant.

[Graphics:../Images/tricks_gr_377.gif]
[Graphics:../Images/tricks_gr_378.gif]

It is straightforward to turn these steps into a [Graphics:../Images/tricks_gr_379.gif].

[Graphics:../Images/tricks_gr_380.gif]

Here is the discriminant for [Graphics:../Images/tricks_gr_381.gif]:

[Graphics:../Images/tricks_gr_382.gif]
[Graphics:../Images/tricks_gr_383.gif]

We can also compute the discriminant using [Graphics:../Images/tricks_gr_384.gif] in the standard package [Graphics:../Images/tricks_gr_385.gif].

[Graphics:../Images/tricks_gr_386.gif]
[Graphics:../Images/tricks_gr_387.gif]
[Graphics:../Images/tricks_gr_388.gif]

Another method is to express the [Graphics:../Images/tricks_gr_389.gif]-th root of the general cubic [Graphics:../Images/tricks_gr_390.gif] as a [Graphics:../Images/tricks_gr_391.gif] object and use [Graphics:../Images/tricks_gr_392.gif].

[Graphics:../Images/tricks_gr_393.gif]
[Graphics:../Images/tricks_gr_394.gif]
[Graphics:../Images/tricks_gr_395.gif]

Document converted by Mathematica of Wolfram Research