Timeline for Bounds on the number of zeros of a quadratic form
Current License: CC BY-SA 3.0
11 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Nov 21, 2015 at 11:28 | comment | added | Keivan Karai | Thanks! I guess I have sorted out the situation using your previous comments. | |
| Nov 20, 2015 at 19:05 | comment | added | Will Jagy | Keivan, I wrote a complete proof and derivation, about 25 pages, for the $2(x^2 + y^2 + z^2) - 113 (yz+zx+xy)= 0.$ If you email me (see profile) I can mail you the (Latex) pdf. Initially, I was not sure what I was looking for, so there is a section involving the adjoints of the Gram matrices that i would probably not do now. Extended notes to myself rather than a polished article. | |
| Nov 20, 2015 at 2:57 | comment | added | Will Jagy | Meanwhile, take a look at mathoverflow.net/questions/208158/isotropic-ternary-forms/… which illustrates the FK language. | |
| Nov 19, 2015 at 23:44 | comment | added | Will Jagy | should not have said integrally equivalent, an isotropic ternary integrally represents a multiple of $y^2 - z x.$ | |
| Nov 19, 2015 at 23:39 | comment | added | Will Jagy | here is FK online, the whole thing. books.google.com/… | |
| Nov 19, 2015 at 23:05 | comment | added | Will Jagy | It's pages 507-508 in FK. The result has been repeated in later summaries. In brief, an indefinite ternary form, integer coefficients, that really is isotropic over $\mathbb Q$ and therefore $\mathbb Q,$ is integrally equivalent to an integer multiple of $y^2 - zx.$ The value of this is that we can write down the primitive null vectors of this latter form, $(u^2, uv, v^2)$ up to $\pm.$ | |
| Nov 19, 2015 at 20:23 | comment | added | Keivan Karai | Thanks for the reference. I mentioned that particular form as an example, but I am indeed interested in dimensions 3 and 4. I will look up Fricke and Klein. | |
| Nov 19, 2015 at 20:16 | comment | added | Will Jagy | Alright. I have an appointment. Dimension 3 is rather special, in that we can parametrize all solutions with a finite number of recipes, similar to the formula for primitive Pythagorean triples; in your example, this follows from simple stereographic projection, in general, it comes from a fact in Fricke and Klein (1897). Meanwhile, dimension 4 has this problem: arxiv.org/abs/1205.4416 | |
| Nov 19, 2015 at 20:02 | comment | added | Keivan Karai | Will Jagy: Here is an example: Take (say) the form $Q(x,y,z)=x^2+y^2-5z^2$. Suppose I want to have solutions $(x,y,z)$ to $Q(x,y,z)=0$ such that $x$ is not divisible by a certain prime number $p$. Is there a guarantee I can get $cT log T$ of them with $|x|,|y|,|z|<T$? | |
| Nov 19, 2015 at 17:23 | comment | added | Will Jagy | I think you had better include some examples with, say, indefinite binary forms. There are some results, heavily dependent on dimension, but I'm not sure what you want to do with the $\pmod m$ or why you want to do so. | |
| Nov 19, 2015 at 10:18 | history | asked | Keivan Karai | CC BY-SA 3.0 |