Editing Bézout's theorem (section)

==History==
In the case of plane curves, Bézout's theorem was essentially stated by [[Isaac Newton]] in his proof of [[Newton's theorem about ovals|Lemma 28]] of volume 1 of his ''[[Philosophiæ Naturalis Principia Mathematica|Principia]]'' in 1687, where he claims that two curves have a number of intersection points given by the product of their degrees.{{sfn|Newton|1966}} However, Newton had stated the theorem as early as 1665.<ref>{{Cite book |last=Stillwell |first=John |url=https://books.google.com/books?id=WNjRrqTm62QC&pg=PA113 |title=Mathematics and Its History |date=1989 |publisher=Springer New York |isbn=978-1-4899-0009-8 |series= Undergraduate Texts in Mathematics|location= |pages=113 |doi=10.1007/978-1-4899-0007-4}}</ref>

The general theorem was later published in 1779 in [[Étienne Bézout]]'s ''Théorie générale des équations algébriques''. He supposed the equations to be "complete", which in modern terminology would translate to [[generic polynomial|generic]]. Since with generic polynomials, there are no points at infinity, and all multiplicities equal one, Bézout's formulation is correct, although his proof does not follow the modern requirements of rigor. This and the fact that the concept of [[intersection multiplicity]] was outside the knowledge of his time led to a sentiment expressed by some authors that his proof was neither correct nor the first proof to be given.<ref>{{cite book | authorlink=Frances Kirwan | last=Kirwan | first=Frances | title=Complex Algebraic Curves | publisher=Cambridge University Press| location=United Kingdom | year=1992 | isbn=0-521-42353-8}}</ref>

The proof of the statement that includes multiplicities requires an accurate definition of the [[intersection multiplicity|intersection multiplicities]], and was therefore not possible before the 20th century. The definitions of multiplicities that was given during the first half of the 20th century involved continuous and infinitesimal [[deformation theory|deformations]]. It follows that the proofs of this period apply only over the field of complex numbers. It is only in 1958 that [[Jean-Pierre Serre]] gave a purely algebraic definition of multiplicities, which led to a proof valid over any algebraically closed field.{{sfn|Serre|1965}}

Modern studies related to Bézout's theorem obtained different upper bounds to system of polynomials by using other properties of the polynomials, such as the [[Bernstein–Kushnirenko theorem]], or generalized it to a large class of functions, such as [[Nash functions]].<ref>{{Cite journal |last=Ramanakoraisina |first=R. |date=1989 |title=Bezout theorem for nash functions |journal=[[Journal of Pure and Applied Algebra]] |language=en |volume=61 |issue=3 |pages=295–301 |doi=10.1016/0022-4049(89)90080-7|doi-access=free }}</ref>