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BÉZOUT'S THEOREM: Let $F$ and $G$ be projective plane curves of degree $m$ and $n$ respectively. Assume $F$ and $G$ have no common component. Then

$\displaystyle\sum_{P}I(P,F\cap G)=mn$

$I(P,F\cap G)$ is the intersection number of $F$ and $G$ at $P$.

I would like to get any reference about the history of this interesting result.

Thank you in advance.

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    $\begingroup$ Hey, this is on the Hot Network Questions page! $\endgroup$ – HDE 226868 Nov 24 '14 at 1:00
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The big surprise I got from my research was that the theorem apparently originated with Maclaurin, whom we remember more for the Maclaurin series than this theorem. From this pdf (you have to go to the last page to get to the history portion):

  1. Maclaurin, Euler, Cramer (1700’s) assert the theorem, no valid proof

But it was Bézout who found a proof, albeit flawed:

  1. Etienne Bezout (1730-1783), flawed proof, didn’t account for multiplicities correctly

Later on, Halphen and van der Waerden came up with accurate proofs that trumped Bézout's.


This biography appears to allude both to Maclaurin's discovery of the theorem and Bézout's "proof":

In this work Bézout also gave the first satisfactory proof of a result of Maclaurin on the intersection of two algebraic curves.

"This work" seems to refer to Théorie générale des équations algébraiques, published in 1779.

I can't seem to find a reference to the theorem on the site's Maclaurin page.


As for specific references, I can only give you these (by the way, Wikipedia cites Math Overflow!):

Wikipedia does say, though,

Bezout's theorem was essentially stated by Isaac Newton in his proof of lemma 28 of volume 1 of his Principia, where he claims that two curves have a number of intersection points given by the product of their degrees. The theorem was later published in 1779 in Étienne Bézout's Théorie générale des équations algébriques. Bézout, who did not have at his disposal modern algebraic notation for equations in several variables, gave a proof based on manipulations with cumbersome algebraic expressions. From the modern point of view, Bézout's treatment was rather heuristic, since he did not formulate the precise conditions for the theorem to hold. This led to a sentiment, expressed by certain authors, that his proof was neither correct nor the first proof to be given.

The source cited is Complex Algebraic Curves, by Frances Kirwan.

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  • $\begingroup$ I'm glad it helped. It was a good question. $\endgroup$ – HDE 226868 Nov 24 '14 at 0:14

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