§ Discriminant and Resultant
I had always seen the definition of a discriminant of a polynomial as:
While it is clear why this tracks if a polynomial has repeated roots
or not, I could never motivate to myself or remember this definition.
I learnt that in fact, this comes from a more general object, the resultant
of two polynomials , which provides a new polynomial
which is zero iff share a common root. Then, the discriminant is
defined as the resultant of a polynomial and its derivative. This makes far more
- If a polynomial has a repeated root , then its factorization will be of the form . The derivative of the polynomial will have an term that can be factored out.
This cleared up a lot of the mystery for me.
- On the contrary, if a polynomial only has a root of degree 1, then the factorization will be , where is not divisible by . Then, the derivative will be . We cannot take common from this, since is not divisible by .
§ How did I run into this? Elimination theory.
I was trying to learn how elimination theory works: Given a variety
, how does one find a rational parametrization
such that , and are
rational functions? That is, how do we find a rational parametrization of the
locus of a polynomial ? The answer is: use resultants!
- We have two univariate polynomials , where the notation means that we have a polynomial . The resultant isa polynomial which is equal to when and share a common root.
- We can use this to eliminate variables. We can treat a bivariate polynomial as a univariate polynomial over the ring . This way, given two bivariate polynomial , , we can compute their resultant, giving us conditions to detect for which values of , there exists a common such that and share a root. If are constants, then we get a polynomial that tracks whether and share a root.
- We can treat the implicit equation above as two equations, , . We can apply the method of resultants to project out from the equations.
§ 5 minute intro to elimination theory.
Recall that when we have a linear system , the system has a non-trivial
solution iff . Formally: . Also, the
ratio of solutions is given by:
If we have two polynomials , and
, then the system , has
a simeltaneous zero iff:
§ Big idea
The matrix is setup in such a way that any solution vector such that
will be of the form . That is,
the solution vector is a polynomial , such that . Since ,
we have that , and .
- Necessity is clear. If we have some non trivial vector such that , then we need .
- Sufficiency : Since , there is some vector such that . We need to show that this is non-trivial. If the polynomials , are not equal, then we have that the rows which have coefficients from and are linearly independent. So, the pair of rows , and the pair are linearly independent. This means that the linear system:
Since the coefficients of the two systems are the same, we must have that
and are linearly dependent. That is:
We can take arbitrarily, giving us a vector of the form
, which is the structure
of the solution we are looking for!