Timeline for Nondegenerate projective plane conics over a nonperfect field $k$ of even characteristics
Current License: CC BY-SA 3.0
9 events
| when toggle format | what | by | license | comment | |
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| Oct 3, 2016 at 12:59 | comment | added | nfdc23 | I am not aware of any other reference discussing his method; it is a pity that modern books omit it. But the relevant parts of Disquisitiones are not so difficult to read as you might fear (if you have't tried). One doesn't need to read all that comes before, and the very detailed table of contents near the front makes it feasible to zero in on the relevant parts. He uses the archaic notation $a R b$ and $a N b$ for respectively saying $a$ is and is not a square (a "residue") modulo $b$, and he uses a $2 \times 3$ matrix notation for binary quadratic forms, but otherwise it is quite readable. | |
| Oct 3, 2016 at 8:46 | comment | added | Dimitri Koshelev | @nfdc23, are there more modern sources about Gauss method? | |
| Oct 3, 2016 at 0:47 | comment | added | nfdc23 | Oops: in my first comment "assing" should be "passing". (Typo or auto-correct? ...) | |
| Oct 2, 2016 at 16:00 | comment | added | nfdc23 | Gauss explains it well in Disquisitiones (and illustrates ideas with very illuminating numerical examples), except that (i) he didn't have the language of linear algebra (so his version of the determinant is negative of ours), (ii) he doesn't point out the geometry behind some formulas (e.g., for binary reduction theory, which he needs for all discriminants for use in the ternary method), and (iii) he only ever considers forms with even cross-term coefficients (sorry!). Due to (iii) it isn't clear if his method adapts to characteristic 2, but it is worth trying. I recommend to read Gauss. | |
| Oct 2, 2016 at 15:25 | comment | added | Dimitri Koshelev | @nfdc23, can you explain more this method? I am actually interested in the global function field $\mathbb{F}_q(t)$ case, where $q = 2^d$, $d \in \mathbb{N}$. | |
| Oct 1, 2016 at 21:08 | comment | added | nfdc23 | Concerning #2, brute force is probably a pain since over $\mathbf{Q}$ that seems to be a rather unpleasant method. But over $\mathbf{Q}$ the Disquisitiones gives a method using as input assing the archimedean and congruential tests at all places away from the 2-adic one: Gauss carried out reduction theory for the binary quadratic parts of the ternary quadratic form (i.e., work on the $xy$-part, then the $yz$-part, then the $xz$-part, and then maybe back to some others, etc.). Maybe that method works over the global function field $\mathbf{F}(t)$ for any finite $\mathbf{F}$ (allowing char. 2)? | |
| Oct 1, 2016 at 16:52 | answer | added | nfdc23 | timeline score: 2 | |
| Oct 1, 2016 at 16:25 | comment | added | abx | 0. By "non-degenerate" you probably mean smooth.$\hskip8.5cm$ 1. Yes. The embedding of $C_i$ in $\mathbb{P}^2$ is given by the anticanonical system $|K_{C_i}^{-1}|$, it is ... canonical.$\hskip2cm$ 2. Brute force doesn't seem too complicated in your case. | |
| Oct 1, 2016 at 14:29 | history | asked | Dimitri Koshelev | CC BY-SA 3.0 |