Thread: Algebraic Geometry - Intersections at the Origin

1. 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!

2. 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.

3. Thanks a lot, you saved me! I completely understand the concept now.