Timeline for Quick proofs of hard theorems
Current License: CC BY-SA 2.5
9 events
| when toggle format | what | by | license | comment | |
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| Dec 3, 2019 at 9:17 | comment | added | Kostya_I | For the record, here is a complete ground up proof (Argan 1809) that fits into an MO comment. Choose a large box such that $|P(z)|>|P(0)|$ on the boundary of the box, and observe that by Bolzano-Weierstrass applied coordinate-vise, $|P(z)|$ attains its minimum inside the box, say at $z_0$. By plugging $z=w+z_0$ into $P$ and expanding, we can rewrite $P(z)=Q(w)=a+bw^k+w^{k+1}R(w)$, where $a$ and $b$ are non-zero and $R$ is a polynomial. Then, put $w:=(a/b)re^{i\pi/k}$ and observe that for a small $r>0$ , $|Q(w)|=|a(1-r^k+O(r^{k+1}))|<|a|=|P(z_0)|$, a contradiction. | |
| Jan 25, 2011 at 0:47 | comment | added | Todd Trimble | The FTA can be proven very quickly and with very little machinery: see ncatlab.org/nlab/show/… | |
| May 17, 2010 at 22:11 | history | edited | gowers | CC BY-SA 2.5 |
Corrected typo
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| May 17, 2010 at 15:12 | comment | added | Andrés E. Caicedo | Ha! @KConrad: I've been looking at the notes at your site. Very nice! | |
| May 17, 2010 at 4:54 | comment | added | KConrad | What I meant by "this (Derksen's proof) is not really a quick proof" is that even if you try to explain it to a mathematician I still think it will take a bit of lead-in to get into the argument and you don't walk away thinking "oh, that was very natural", which was the attitude which the original question was about. By the way, are you saying Derksen's proof is the fastest you know which starts from scratch? There are proofs by multivariable calculus which take less time. A proof with double integrals is at math.uconn.edu/~kconrad/blurbs/fundthmalg/… | |
| May 17, 2010 at 1:08 | comment | added | Steven Gubkin | What about the topological proof? Building up to the fundamental group of the circle only takes about a page! | |
| May 17, 2010 at 0:09 | comment | added | Andrés E. Caicedo | " Well, this is not really a quick proof. The proofs by complex variables are much faster (granting enough build-up in complex variables) " Oh, sure, "granting enough build-up" is the key here. This proof is perhaps the fastest I know from the ground up. But we rarely start at the ground anymore. | |
| May 16, 2010 at 23:48 | comment | added | KConrad | Well, this is not really a quick proof. The proofs by complex variables are much faster (granting enough build-up in complex variables). To be fair, a few years ago I made Derksen's proof the goal of my undergraduate linear algebra course for math majors, but it took me two days to go through the argument carefully and I decided it might have been hard for them to appreciate when the argument goes on that long. | |
| May 16, 2010 at 22:57 | history | answered | Andrés E. Caicedo | CC BY-SA 2.5 |