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Here KReiser showed that a distinct linear factor of the resultant $R_{P,Q}$ can correspond to more than one point of intersection of the projective curves $P(x,y,z)=0,Q(x,y,z)=0$.

I've been reading Kirwan's Complex Algebraic Curves for exposition on this topic; in it, he defines the intersection multiplicity axiomatically. He shows that the multiplicity of the linear factor in the resultant corresponding to the intersection point satisfies these axioms.

However, how can this reconcile with Bézout's Theorem? Kirwan's proof explicitly uses the fact that each factor corresponds to a unique point: enter image description here

jonathan
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    I should point out that the answer of mine you linked shows that if there are $n$ intersection points on the line $(ay-bz)$, then $(ay-bz)$ shows up with degree $n$ in the resultant, so it is still true that linear factors correspond in a 1-1 fashion to points of intersection. – KReiser Nov 24 '19 at 22:02
  • @KReiser Thats why I'm confused. Is there a mistake in Kirwan's book? – jonathan Nov 24 '19 at 22:13
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    I can't tell because there's a reference to stated assumptions in this proof which aren't on the page you provided. If, for instance, you enforce the condition from the comment on your last question you do get what you want. – KReiser Nov 24 '19 at 22:27

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