# How would have Bezout proved Bezout's theorem?

How would Bezout have proved Bezout's theorem bounding the number of points in the intersection of two plane (polynomial) curves in $$\mathbb{R}^2$$?

I have looked at a couple of modern algebraic treatments of Bezout's theorem. For instance, I am familiar with Fulton's Algebraic Curves and his treatment of Bezout's theorem therein. While a purely algebraic approach is nice and has its uses, I feel like that proof and related modern proofs must be very different from Bezout's. I expect that Bezout's proof could not have used much beyond rudimentary calculus; in particular, I doubt he would have used the projective plane, exact sequences or local rings. Given that he was an 18th century mathematician, it is also unclear to me if he could even have used the Fundamental Theorem of Algebra which makes me uncertain that he would have proved it using resultants e.g., some version of a proof outlined on the wikipedia page: https://en.wikipedia.org/wiki/B%C3%A9zout%27s_theorem .

Alternatively, my question is what is the most classically analytic or down-to-earth proof of Bezout's theorem? Regarding analytic approaches to Bezout's theorem, I think that Griffiths--Harris proof (around page 171 therein) is analytic, but not classical.

Bonus points: If such a proof is substantially different from that in Fulton's Algebraic Curves, then can you describe how the proofs relate (assuming that they do) or how the classical proof inspires the modern proof?

• The proof appears in Bézout's own book Théorie générale des équations algébriques (1779). A 2002 English translation by Eric Feron is available -- the theorem is presented in paragraph 47, which is on page 24 of the translation. Have you read this? It seems like a natural place to start... Aug 18, 2021 at 12:36
• Also, the statement "I doubt he would have used [...] exact sequences or local rings" has my vote for the understatement of the century (or two centuries, to be exact) :-) Aug 18, 2021 at 12:37
• The Fundamental Theorem of Algebra can be considered as a 1-dimensional version of Bezout, and it is absolutely essential. Without it, you may only get an inequality. Resultants and elimination theory is a natural and classical approach. Aug 18, 2021 at 12:40
• @Carl-FredrikNybergBrodda Thank you for that reference! Unfortunately, I cannot access the book without paying for it. However, searching for an arxiv version led me to the following reference which appears helpful: arxiv.org/pdf/1606.03711.pdf Aug 18, 2021 at 13:21
• drive.google.com/file/d/1cYPvDagNHsM39ngUjr--HftY44Zznmz5/view Bezout proof was not rigorous. A simple proof can be obtained using resultants. Aug 18, 2021 at 16:27