Timeline for Quick proofs of hard theorems
Current License: CC BY-SA 4.0
11 events
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| Feb 7, 2020 at 15:30 | history | edited | Martin Sleziak | CC BY-SA 4.0 |
http -> https (the question has been bumped anyway)
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| May 18, 2010 at 2:22 | comment | added | Victor Protsak | The "old proof" refers to the finite generation of the ring of invariants (for the case of SL(2) acting of binary forms), which is a different theorem. Hilbert found an encompassing approach based on Noether condition for ideals (in fact, he proved much more, in particular, the finiteness of the chain of syzygies); note, however, that in order to apply his "theological" proof to the original problem for more general invariant rings, one still needs to know complete reducibility of representations, which was first established by Hurwitz and Schur using topological arguments (compactness). | |
| May 17, 2010 at 13:39 | comment | added | David Corfield | Do read McLarty's article people.math.jussieu.fr/~harris/theology.pdf on Gordan's attitude towards Hilbert's work. Historical reality is much more interesting than the myth. | |
| May 16, 2010 at 23:52 | comment | added | Mariano Suárez-Álvarez | The quote is Gordan's. | |
| May 16, 2010 at 23:29 | comment | added | Qiaochu Yuan | @Harry: ah, that makes sense. What I think I remember reading was that it had previously only been proven for polynomial rings over fields by explicit computation. | |
| May 16, 2010 at 23:23 | comment | added | Steve D | This is, I believe, also the source of Gordon's famous "theology" quote. | |
| May 16, 2010 at 22:50 | comment | added | Harry Gindi | @Qiaochu: I read an article on the history of the HBT, and the way it actually "went down" was that the theorem was proven individually for explicit rings. Hilbert proved the HBT and in doing so proved the general theorem. | |
| May 16, 2010 at 20:37 | comment | added | darij grinberg | Okay, that's always the question with the word "constructive". If we have an ideal given by some equations, or even by generators, how much do we know about the set of leading coefficients of elements of this ideal? Not enough to find its generators. But then again, Hilbert's basis theorem is not 100% constructive itself, for this very reason: we have no idea how the ideal is given. | |
| May 16, 2010 at 20:23 | comment | added | Qiaochu Yuan | Well, there is a proof labeled "a constructive proof" in the Wikipedia article... | |
| May 16, 2010 at 20:16 | comment | added | darij grinberg | What? I think there is by definition no constructive proof of Hilbert's basis theorem. Maybe you mean the Gröbner basis for the invariant ring of a group action, or the projective resolution? These are both still best proven constructively, as the constructive proofs yield tons of additional results. | |
| May 16, 2010 at 19:40 | history | answered | Qiaochu Yuan | CC BY-SA 2.5 |