The discriminant Δ of an -th degree polynomial with roots is the symmetric product . (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 (implicitly via an order term ) and . Consider the case .
With the Groebner basis for the coefficients,
we can use to express the discriminant in terms of the coefficients β and γ.
The last term of this expression is the discriminant.
It is straightforward to turn these steps into a .
Here is the discriminant for :
We can also compute the discriminant using in the standard package .
Another method is to express the -th root of the general cubic as a object and use .
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