Timeline for On the equation $q(\mathbf{x}) = 1$ for $q$ a quadratic form
Current License: CC BY-SA 4.0
8 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Dec 9, 2023 at 21:02 | comment | added | Will Jagy | and pages 301-309 in Cassels, Rational Quadratic Form s | |
| Dec 9, 2023 at 18:21 | comment | added | Will Jagy | in dimension 3 the integer automorphism group of the form can be parametrized by $SL_2 \mathbb Z$ This goes back to Fricke and Klein (1897) and is displayed in Magnus, Noneuclidean Tesselations. Ummmm. The kids like parametrizations..., two I did in November..... math.stackexchange.com/questions/4802380/… math.stackexchange.com/questions/3221394/… | |
| Dec 8, 2023 at 23:55 | comment | added | Bogdan Grechuk | For a nice example of equation of your type, see mathoverflow.net/questions/432091 | |
| Dec 8, 2023 at 23:54 | comment | added | Bogdan Grechuk | See my answer to this question mathoverflow.net/questions/142938 about general quadratic equations. | |
| Dec 8, 2023 at 21:03 | comment | added | GH from MO | I am sure you are aware of this, but the solutions of $q(x_1,\dotsc,x_n)=1$ form finitely many orbits under the group of automorphs of $q$, and a solution exists if and only if $q$ is properly equivalent to a form $x_1^2+r(x_2,\dotsc,x_n)$. | |
| Dec 8, 2023 at 18:53 | comment | added | Stanley Yao Xiao | @LSpice certainly Hasse's principle on rational solutions would be too weak (i.e., that there is a rational solution if and only if the equation has a local solution in every completion of $\mathbb{Q}$). I am mostly thinking about the case when there are infinitely many solutions, and what can be said about the set of solutions. | |
| Dec 8, 2023 at 18:44 | comment | added | LSpice | Do you have any sort of criteria for what would constitute a reasonable decription? For example, would an emptiness vs. non-emptiness, or finite vs. infinite, classification be reasonable, or too weak? (I don't know the answers even to these special cases, just am curious about what sort of answer is sought.) | |
| Dec 8, 2023 at 18:05 | history | asked | Stanley Yao Xiao | CC BY-SA 4.0 |