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I have only just come across the remarkable theorem of Conway about universal quadratic forms over $\mathbb{Z}$; namely that in determining whether a integer coefficient, positive definite quadratic form represents all positive integers it suffices to check that it represents a specific set of 29 integers (the largest of which is 290). Other similar theorems exist too about representing all primes or all odd numbers.

I just read the paper by Bhargava and Hanke and found the proof quite elegant. I got thinking about whether it would generalize to other situations.

Has anyone has been able to extend the results to other settings? Maybe people have been able to prove similar things over other rings (such as rings of integers of number fields) or maybe people are still considering staying in the integer case and considering representing other sets of integers or considering higher degree forms?

fretty
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  • Could you give references or links to these papers? – lhf Apr 20 '13 at 14:23
  • I am on my phone at the moment but a google search for "290 theorem Bhargava" will get it. – fretty Apr 20 '13 at 14:36
  • @lhf, I have a pdf of Quadratic Forms and Their Applications, edited by Eva Bayer-Fluckiger, David Lewis, and Andrew Ranicki. This has Conway's introductory article and Bhargava's proof of the 15 theorem. I know a friend of Hanke sent me the article part of the 290 theorem, if I can find it. This was accepted by Inventiones but withdrawn after Hanke concluded that he could greatly improve the methodology. He has now left academia, it is unlikely the article (or list of 6000+ forms) will appear. – Will Jagy Apr 20 '13 at 14:37
  • @lhf, I already had Bhargava's 15 article at http://zakuski.math.utsa.edu/~kap/forms.html under the name Bhargava_2000 as a pdf. If I can remember how to do it, I will put in the Bhargava-Hanke 290 preprint. – Will Jagy Apr 20 '13 at 14:45
  • Ok but I was wondering about generalizations of this result. Has anyone been able to provide similar results in other settings or for higher degree forms? – fretty Apr 20 '13 at 15:32
  • fretty, if you want to know more than my answer below, it should probably be by email. Meanwhile, I do not know of anyone succeeding in enumerating all universal, or positive universal forms, in integer rings of number fields or with cubic or quartic or higher forms. I might not know, of course. My guess is the problems only get harder. – Will Jagy Apr 20 '13 at 15:38
  • Oh sorry, didn't see your answer! My phone is playing up... – fretty Apr 20 '13 at 15:40

2 Answers2

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Quite a bit of information is available as pdfs at my page TERNARY with what I hope are obvious names.

You want to look at Rouse on all odd numbers, ROUSE. Also Representation by ternary quadratic forms by OLIVER. In both cases some ineffective bounds are used, so a GRH is invoked that implies the suspected conclusions. This gives about the best conclusion to my paper with Kaplansky and Schiemann that I have any right to expect.

Hanke certainly thought that almost anything could be extended to integer rings of some number fields, and intended to find all class number one genera. This was an ambitious project, as it would require dimension up to 26. A student of Gabriele Nebe, named David Lorch, has found all positive class number one forms over $\mathbb Z,$ see LORCH.

I do not believe I know of any big-list papers on universality over number rings. There are some related approaches by Pete L. Cark of MO and MSE, see item 15 at CLARK. In this case, there was surely some influence by Hanke, who was at Georgia for some years.

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Will Jagy
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  • I like the general theorem of Bhargava, that any subset $S$ of the natural numbers has a unique finite subset $S0$ that measures representability by everything in $S$. From what I gather though there has not been much discovered in other rings then? – fretty Apr 21 '13 at 09:02
  • @fretty, the history of that is more Conway. When he proposed the problem in this manner, the first thing to happen was the proof that there was some constant that sufficed, which turned out to be 15 for "integer-matrix" forms and 290 for unrestricted positive. You might try to prove that there is such a constant for some ring you have in mind; the specific small subset comes later. Note that this all is very, very different for indefinite $\mathbb Z$ forms, there are infinitely many indefinite ternaries. Bear that in mind for other rings. – Will Jagy Apr 21 '13 at 21:42
  • @fretty, sorry, infinitely many universal indefinite ternaries, $xy + 17 z^2$ or anything you like instead of 17. There could be indefinite ternaries of prime discriminant $q$ that represent all numbers, positvie and negative, not divisible by $q,$ but only a portion of multiples of $q.$ So, maybe there is no such bound without an ordered ring and positivity. – Will Jagy Apr 21 '13 at 22:14
  • A shame that the paper may not appear... – Andrés E. Caicedo Apr 21 '13 at 22:46
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    @AndresCaicedo, well, Bhargava keeps talking about it, so anything is possible. Plus, Hanke does want to keep one foot, or one toe, in academia, he has said he will do some computations for Wai Kiu Chan of Wesleyan. These big-list papers are a bit iffy. A ton of work, not elegant once you get to the list itself, and noone else will ever really understand what struggles went into the computations. – Will Jagy Apr 21 '13 at 22:57
  • I find the argument quite elegant and I believe there must be some way of generalizing the ideas to other settings...I agree that some kind of ordering/positivity will be needed (maybe representation of totally positive elements). – fretty Apr 22 '13 at 08:58
  • (An update.) – Andrés E. Caicedo Jul 20 '13 at 01:27
  • @AndresCaicedo, thanks. – Will Jagy Jul 20 '13 at 01:46
  • The first link seems to be broken, but http://zakuski.math.utsa.edu/~kap/ works fine. – Watson Feb 10 '18 at 19:20
  • @Watson fixed as you indicate. A few years ago, the host server died; a year later, a portion of the content was put into a revised directory structure. Not sure I had noticed that Hanke had put an answer here. My page of pdfs is focused on positive ternary quadratic forms, while the 290 material leads naturally to larger dimension – Will Jagy Feb 10 '18 at 19:34
  • @WillJagy mate, can you please tell when is a quadratic form ternary? –  Feb 07 '22 at 08:09
  • The Oliver link is broken. The paper is now at http://www.mathcs.emory.edu/~rlemkeo/papers/09-TernaryQuadratics.pdf – Rosie F May 30 '22 at 13:53
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Here is my webpage with information about the 290-Theorem. It has links to the preprint, all escalator form datafiles and computer code. In Spring 2014 I had a student (Kate Thompson) graduate from UGA after learning about the analytic techniques involved in doing similar computations for totally positive definite $\mathcal{O}_F$-valued quadratic forms over $\mathcal{O}_F := \mathbb{Z}\left[\frac{1 + \sqrt{5}}{2}\right]$. If you're interested in learning more about the theory behind this theorem, here is a video of the talk I gave at Rutgers last year about it.

It's certainly possible to prove a version of the 290-Theorem to establish universality of forms over the ring of integers $\mathcal{O}_F$ of an arbitrary totally real number field $F$ (modulo some conditions about no ternary escalator forms appearing), but it is a highly non-trivial task to create a general code base to prove what numbers are represented by an arbitrary totally definite $\mathcal{O}_F$-valued quadratic form in 4 variables. This is what was done over $\mathbb{Z}$ for the 290-Theorem, and then was used on 6664 quadratic forms to establish the result. The point is that doing it without a computer is not feasible, but writing a program to accurately perform this task is very difficult and very time consuming (and may adversely affect someone's career unless they have tenure). To give you a sense of it -- this project took me about 4 years of focused work to initially complete the algorithm development and coding/debugging, and several more years to recheck the computations to my satisfaction.

I'm sure that the paper will appear as more than a preprint at some point, and it's certainly not something that I've forgotten about. =)

Jonathan Hanke
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  • This is not about solving quadratic forms but representability of a given quadratic form... – fretty Aug 14 '14 at 08:46
  • Many thanks for the reply Jonathan! So I guess the path is clear but the computational results are either not known or are far more demanding? – fretty Aug 14 '14 at 09:52