Results 1 to 3 of 3

Math Help - Algebraic Geometry - Intersections at the Origin

  1. #1
    Newbie
    Joined
    Apr 2008
    Posts
    17

    Algebraic Geometry - Intersections at the Origin

    I'm clueless as to how to even start one of these problems in the graphic below:



    If somebody could please tell me how to do one of them, it would help so much. I was fine when asked to calculate the intersection at the origin of two specified curves, but I really don't know how to find curves that intersect a given curve a given amount of times. I'm just totally and utterly confused on it. Thanks!
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor
    Opalg's Avatar
    Joined
    Aug 2007
    From
    Leeds, UK
    Posts
    4,041
    Thanks
    7
    A line through the origin has equation y = kx, unless it is the vertical line through the origin, which has equation x = 0.

    To find how many times the line y = kx intersects the curve f(x,y) = 0 at the origin, put y = kx and find the multiplicity of the root x=0 in the equation f(x,kx) = 0. To find how many times the line x = 0 intersects the curve f(x,y) = 0 at the origin, put x=0 and find the multiplicity of the root y=0 in the equation f(0,y) = 0.

    As an example look at problem (j) in that list. We want to show that there are two lines that intersect the curve (x^2+y^2)^2=xy^2 more than three times at the origin, and that all other lines through the origin intersect the curve exactly three times there. So put y=kx in the equation of the curve. You get x^4(1+k^2)^2=k^2x^3. In this equation, x=0 is a triple root for all values of k except k=0, when it becomes a quadruple root. So apart from the single value k=0, you get a line that meets the curve three times at the origin. So far, we only have one exceptional value. But now look at the vertical line. Put x=0 in the equation of the curve, and it becomes y^4=0. Here, y=0 is a quadruple root, so that line meets the curve four times at the origin.
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Newbie
    Joined
    Apr 2008
    Posts
    17

    Thumbs up

    Thanks a lot, you saved me! I completely understand the concept now.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. algebraic geometry
    Posted in the Advanced Algebra Forum
    Replies: 10
    Last Post: October 22nd 2011, 07:13 PM
  2. algebraic geometry
    Posted in the Advanced Math Topics Forum
    Replies: 0
    Last Post: November 11th 2009, 03:24 PM
  3. algebraic geometry
    Posted in the Advanced Math Topics Forum
    Replies: 0
    Last Post: April 6th 2009, 07:32 AM
  4. Algebraic Geometry
    Posted in the Advanced Algebra Forum
    Replies: 1
    Last Post: October 23rd 2008, 07:49 AM
  5. Algebraic Geometry
    Posted in the Advanced Algebra Forum
    Replies: 1
    Last Post: October 19th 2008, 09:37 PM

Search Tags


/mathhelpforum @mathhelpforum