Bézout’s theorem

  • P. B. Kronheimer
  • Published 2007


This handout gives a statement and proof of Bézout’s theorem concerning the intersecion of two plane curves. The background to the proof is: (i) the definition of intersection multiplicity which we gave in class and which is presented in a separate handout; and (ii) the basic facts about resultants. The presentation here is rather different from that in Kirwan’s book, because our definition of intersection multiplicity has a different starting point. (Kirwan defines intersectionmultiplicity using resultants.) The price to be paid is a longer proof, with more abstract algebra. On the other hand, we avoid some awkward questions about the independence of the choice of coordinates that arise when using resultants as Kirwan does. We also deviate from Kirwan’s presentation in that we present Bézout’s theorem in the context of affine plane curves (that is, curves in C2) rather than curves in the projective plane.

Cite this paper

@inproceedings{Kronheimer2007BezoutsT, title={Bézout’s theorem}, author={P. B. Kronheimer}, year={2007} }