Timeline for How to determine the degree of a rational function field over a relatively algebraic subfield?
Current License: CC BY-SA 4.0
9 events
| when toggle format | what | by | license | comment | |
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| Mar 22, 2022 at 18:00 | answer | added | pinaki | timeline score: 2 | |
| Mar 22, 2022 at 12:53 | comment | added | GiS | I think this may provide some help: mathoverflow.net/questions/66427/… | |
| Mar 22, 2022 at 12:14 | history | edited | GiS | CC BY-SA 4.0 |
Missing some important assumption
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| Mar 22, 2022 at 12:13 | comment | added | GiS | I find a counterexample when $n=2$ and $[K(x_1,x_2):K(x_1,x_1x_2)]=1$. Hence I furthermore require $(f_1,\cdots,f_n)$ is a regular sequence. | |
| Mar 22, 2022 at 4:22 | comment | added | GiS | Dear van Dobben de Bruyn, I think you indeed give a counterexample when some $f_i$ are inhomogeneous, that's why I assume that $f_i$ are homogeous. Besides, I guess that my problem is perhaps about projective geometry, and your example is about affine geometry. | |
| Mar 22, 2022 at 4:19 | comment | added | GiS | Dear Sawin, can you give some references about your claim? Thanks. Besides, I think my problem is about Bezout's theorem. | |
| Mar 21, 2022 at 20:29 | comment | added | Will Sawin | This is the number of roots of $f_1=y_1,\dots, f_n=y_n$ in variables $x_1,\dots, x_n$ over the field $\overline{K(y_1,\dots,y_n)}$. As a computational problem, this is surely solvable - I'm going to take a wild guess that the answer involves Grobner bases. | |
| Mar 21, 2022 at 18:24 | comment | added | R. van Dobben de Bruyn | This is definitely not true as stated. Even the question of when $K(f_1,\ldots,f_n)$ is equal to $K(x_1,\ldots,x_n)$ is a subtle one; in characteristic $0$ there is for instance the Jacobian conjecture. For example, $f_1 = x_1+y_2^{100}$ and $f_2 = y_2$ generate $K(x_1,x_2)$, but $d_1d_2 > 1$. | |
| Mar 21, 2022 at 13:32 | history | asked | GiS | CC BY-SA 4.0 |