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As the picture shows2(the paper's link is in 1),it seems that I can use tools including Bezout's theorem to solve the EXACT number of intersection between two algebraic curves(F(x,y) is of degree two and with variables x and y.G(x,y) is of degree four and with variables x,y ).

But so far I don't see it. So please help me if you have any thought on this,thanks!

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  • $\begingroup$ It depends on whether you take into account the multiplicities of intersection or not. Assuming you just want the cardinality of the set-theoretic intersection, this is done in the following nicely written paper: arxiv.org/abs/0907.0361 This has been implemented in Magma: magma.maths.usyd.edu.au/magma/handbook/text/1386#15457 (you can use Magma with the online calculator). $\endgroup$
    – François Brunault
    Commented Jan 15, 2021 at 17:04
  • $\begingroup$ Also, what is your field $k$? Do you mean the $k$-rational points? Do you include the points at infinity? $\endgroup$
    – François Brunault
    Commented Jan 15, 2021 at 18:06
  • $\begingroup$ @FrançoisBrunault Thanks for your help!To be simple I just want to know how many common points do Three ellipses have on the R^2 plane.And that's equivalent to how many common points do two algebraic curves have.I would read the paper through and try Magma as soon as possible after I finish my homework before the deadline.Thanks again! $\endgroup$
    – BobSS
    Commented Jan 16, 2021 at 4:04
  • $\begingroup$ This is not equivalent: Bézout's theorem only gives the number of points over the complex numbers, not the reals. To get the number of real intersection points, you need to use Hilmar-Smyth and count the number of real embeddings of the number fields that appear. $\endgroup$
    – François Brunault
    Commented Jan 16, 2021 at 8:34

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