Discriminant of bivariate (or trivariate) polynomials

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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







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    up vote
    1
    down vote

    favorite












    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







    share|cite|improve this question
























      up vote
      1
      down vote

      favorite









      up vote
      1
      down vote

      favorite











      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







      share|cite|improve this question














      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









      share|cite|improve this question













      share|cite|improve this question




      share|cite|improve this question








      edited Aug 24 at 4:39









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      asked Mar 25 '17 at 12:50









      Shalec

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