Curves intersecting at the origin

In summary: I don't know if there is an easy proof, but I guess you could prove it by induction on the degree of p(x).In summary, the conversation discusses finding the number of times two curves intersect at the origin. One person suggests using a theorem that states the intersection multiplicity at the origin is the smallest degree of any non-zero term in the equation involving both curves. Another person suggests using polar coordinates to find the number of crossings, but it is clarified that this is not the same as the intersection multiplicity.
  • #1
b0mb0nika
37
0
i have to show how many times the curves intersect at the origin

y^4 = x^ 3 and x^2y^3 - y^2+ 2x^7= 0

i don't really know how to start solving this :rolleyes:
 
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  • #2
i think the answer is 6
could anyone tell me if it's right?
 
  • #3
It looks right. You found 22 solutions (over the complex numbers) with x and y non-zero? No solutions for the points at infinity in projective space?
 
  • #4
I plotted them in surf. at the origin one is a U but turned on it's side with the "opening" to the right. the other is a cusp, or locally < shaped and the point of the < meets the rounded bottom of the U, I don't know how many "crossings" you'd wish to count that as.
 
  • #5
shmoe.. i didnt actually find the solutions.
i let t^3=y from the frist equation
and then expressed everything in terms of t in the second equation, and then by
a theorem ( which i don't know the name of).. the lowest power of the non-zero
terms is the number of time the curves intersect at 0.
 
  • #6
Hmm, I'm not sure of the theorem you're using, could you give it's statement or a reference to where it can be found? I'd like to have a look. My idea was to use Bezout's theorem to count the total number of intersections (28). You can replace the 2x^7 with 2x^4y^4, making the second equation a quadratic in the variable x^2. You can then find all non-zero solutions in complex projective space (22 in total) and show the curves intersect only once at each of these, so the origin would have 6 intersections. I admit to being pretty ignorant on most things algebraic though, so I could be way off.
 
  • #7
ok this is how the theorem goes:

let y = p(x) and g(x,y) = 0 be 2 curves. Assume y = p(x) contains 0= (0,0) and that (y-p(x)) does not divide g(x). Then the intersection multiplicity at 0( i assume I_0 .. I sub zero...means that) of y- p(x) and g(x,y) is the smallest degree of any non zero term of g(x,p(x)) .

i'm not sure where this came from, I missed the day when it was thought in class so I got the notes from somebody and I found it there. I don't think its in the book that we use right now for the course cause I was looking for it.
 
  • #8
How about writing them in polar coordinates?
y^4 = x^ 3 becomes r4sin4&theta;= r3cps&theta; or r sin4&theta;= cos3&theta;. Taking r= 0 we have cos&theta;= 0, [itex]\theta= \frac{\pi}{2}[/itex] and [itex]\fra{3\pi}{2}[/itex].
The graph crosses through the origin twice.

x^2y^3 - y^2+ 2x^7= 0 becomes r5cos2&theta;sin3&theta;- r2sin3&theta;+ r7cos7&theta;= 0 or r3(cos2&theta;+ r2cos7&theta;)- sin3&theta;= 0. Taking r= 0, sin3&theta;= 0 so
sin&theta;= 0. [itex]\theta= 0[/itex] or [itex]\theta= \pi[/itex] or [itex]\theta= 2\pi[/itex]. The graph crosses the origin 3 times (notice that 0 and [itex]2\pi[/itex] are different!).
 
  • #9
I don't think that works, the number of "crossings" is not the same as the intersection multiplicity. y^4=x^3 has a zero of multiplicity 3 at the origin, the other graph a zero of order 2 (the multiplicity of the zero is the lowest power term).

b0mb0nika said:
let y = p(x) and g(x,y) = 0 be 2 curves. Assume y = p(x) contains 0= (0,0) and that (y-p(x)) does not divide g(x). Then the intersection multiplicity at 0( i assume I_0 .. I sub zero...means that) of y- p(x) and g(x,y) is the smallest degree of any non zero term of g(x,p(x)) .

Sorry I forgot about this post. This makes sense, I've seen this before in the restricted case of p(x) being linear.
 

Related to Curves intersecting at the origin

What does it mean for curves to intersect at the origin?

When two curves intersect at the origin, it means that the point (0,0) lies on both curves. This is also known as the point of intersection or the point where the two curves meet.

How can I determine if two curves intersect at the origin?

To determine if two curves intersect at the origin, you can graph the two curves and see if they both pass through the point (0,0). Another way is to set the equations of the two curves equal to each other and solve for the variable to see if the solution is (0,0).

What does it signify when two curves intersect at the origin?

When two curves intersect at the origin, it signifies that the two curves have at least one point in common. This point of intersection can have different meanings depending on the context of the two curves.

Do all curves intersect at the origin?

No, not all curves intersect at the origin. Some curves may never pass through the origin, while others may intersect at multiple points. It depends on the equations and the shapes of the curves.

Can two curves intersect at the origin more than once?

Yes, two curves can intersect at the origin more than once. This means that there are multiple points where the two curves meet at the point (0,0). These points of intersection can have different meanings and can also provide multiple solutions to a system of equations.

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