I am wondering if there is any Definition of a discriminant of bivariate or trivariate polynomials?
Lets say you have an arbitrary bivariate polynomial of highest degree 3. (hasn't to be homogenous) Then a homogenization leads to a homogeneous trivariate polynomial. Is there any Definition for a discriminant of those? Or how can I calculate it?
In fact I know the definition of a discriminant for arbitrary polynomials (Cohen, A Course in Computation Algebraic Number Theory, Chapter 3 and 4) and I would like to extend it to the upper behaviour. As [Wikipedia] says, there is a solution for homogeneous bivariate polynomials. Is there something similar to homogeneous trivariate polynomials?
Thank you.
PS: As I tagged "elliptic-curves" there is a solution in this field of investigation. But is there a link to the presented definition by Cohen?
[Wikipedia] https://en.wikipedia.org/wiki/Discriminant#Homogeneous_bivariate_polynomial