Timeline for How would have Bezout proved Bezout's theorem?
Current License: CC BY-SA 4.0
13 events
| when toggle format | what | by | license | comment | |
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| Aug 20, 2021 at 12:29 | comment | added | Carl-Fredrik Nyberg Brodda | @KHughes If you write to me, I can send you a pdf of the English translation of Bézout's book. | |
| Aug 20, 2021 at 12:14 | comment | added | K Hughes | @Carl-FredrikNybergBrodda Thank you for pointing that out. The corrected link to Bezout's book (in French) is: gallica.bnf.fr/ark:/12148/bpt6k106053p.image I am deleting my comment to get rid of the personal link and replacing my comment here: I don't know any French, but from what I can tell Bezout proves a result about resultants which implies that the "Bezout bound" is true generically. | |
| Aug 19, 2021 at 11:32 | comment | added | Carl-Fredrik Nyberg Brodda | @KHughes That's a link to a file on your computer, not to a website. | |
| Aug 19, 2021 at 11:13 | comment | added | K Hughes | To summarize my understanding: From references in these comments and Wikipedia, it appears that the bound in $\mathbb{R}^2$ was even "classical" for Bezout, being attributed to Newton. (The wikipedia reference to Newton's "Lemma 28" seems to be incorrect here, or at least I do not see the connection.) So, my original question should be for higher dimensions where the answer appears to be that Bezout worked with equations whose coefficients were indeterminates to work generically and he brute forced the problem by a case analysis of resultants. I am unsure how much calculus he used. | |
| Aug 19, 2021 at 11:01 | comment | added | K Hughes | @AlexandreEremenko Thank you for the reference. The reference is precisely what I am not looking for because it justifies the modern formulation and proof of Bezout's theorem. I am happy with a non-rigorous proof since most proofs from that time are now regarded as non-rigorous. | |
| Aug 18, 2021 at 16:27 | comment | added | Alexandre Eremenko | 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 14:43 | comment | added | Carl-Fredrik Nyberg Brodda | @KHughes The book can easily be accessed through various standard pseudolegal sources (e.g. at Library בְּרֵאשִׁית, or rather the Greek equivalent). | |
| Aug 18, 2021 at 13:24 | comment | added | K Hughes | @OlegEroshkin I think the original versions of Bezout's theorem were only inequalities which would be in line with not knowing the Fundamental Theorem of Algebra. I agree resultants and elimination theory is the most likely approach for Bezout. Certainly, many people in the 18th century were occupied with that topic. | |
| Aug 18, 2021 at 13:21 | comment | added | K Hughes | @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 12:40 | comment | added | Oleg Eroshkin | 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 12:37 | comment | added | Carl-Fredrik Nyberg Brodda | 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 12:36 | comment | added | Carl-Fredrik Nyberg Brodda | 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 12:22 | history | asked | K Hughes | CC BY-SA 4.0 |