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The sketch begins by rewriting the 2 curves, X and Y, and somehow they're now inexplicably univariate polynomials in terms of z when they where in terms of x and y. If they're now in homogeneous coordinates then they should be in terms of x, y, and z. This, however, seems incompatible with the rest
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I see where my confusion was (what looked like coefficients are actually polynomials in which the x's and y's are hiding. This was stated but I misunderstood it.) and I will add the example I was in need of. Once I understand the proof and have clarified any other stumbling blocks I run into I'll
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of the proof. Can someone add an example (or a link to the missing background material) for how this polynomial rewriting is supposed to happen such that x and y are no longer present, or, if such a rewriting does not exist, provide a real proof?
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The text begins with a somewhat vague definition of the theorem in mathematics in general, and then only in algebraic geometry. I don't understand this distinction; to me, the theorem is really about algebraic geometry.
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Yo, one of the examples is wrong (the one w/ the longer ellipse). That has an intersection of multiplicity 2 at (-1,0), one of multiplicity 1 at (0,1), and one of multiplicity one at (17/8,-15/8*I).
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What about the example with two circles centered at the origin with radius 1 and 2 respectively? I'm getting 0 intersection points instead of 4, how is this resolved?
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Bezout's is a fundamental theorem in projective geometry, although any study of algebraic geometry requires it (since we do algebra on projective varieties mostly).
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I agree that to make it useful much more detail should be added, even if it's just a sketch. As of now I can't make sense of it
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I agree that the first paragraph of the lead was confusing, and I have edit it for clarification.
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clearer: at the moment, it's very imprecise, and confusing. -- Anon.
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