Timeline for Concrete bounds on the discriminant of a number field
Current License: CC BY-SA 3.0
7 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Oct 6, 2016 at 15:10 | comment | added | Ariyan Javanpeykar | Sorry. I somehow misread that part of your question. | |
| Oct 6, 2016 at 13:09 | answer | added | user1073 | timeline score: 9 | |
| Oct 6, 2016 at 12:51 | comment | added | Pablo | @AriyanJavanpeykar thanks for pointing this out, that's cool! Actually (as you can see in my question) I am mostly interested in the non-totally real case. | |
| Oct 6, 2016 at 12:45 | comment | added | Ariyan Javanpeykar | Have a look at Prop. 2.3 in Takeuchi's paper. He proves $d(k) > a^n \exp(-b)$, where $a = 29.099$ and $b=8.3185$. (Here $n$ is the degree of $k$ and $d(k)$ is the discriminant of the totally real number field $k$.) | |
| Oct 6, 2016 at 12:27 | comment | added | Pablo | @AriyanJavanpeykar You are right, but the bounds he gives involve some zeta values, so in particular they depend not only on $n$ (or the signature). This makes it impossible for me to effectively bound the discriminant of a general degree $7$ number field. | |
| Oct 6, 2016 at 12:18 | comment | added | Ariyan Javanpeykar | Dear @Pablo, I might be misremembering, but I think Odlyzko proved an explicit lower bound for the root discriminant of a number field (i.e., not just an asymptotic result). See Theorem 1.(2) in Lower bounds for discriminants of number fields. II. Tôhoku Math. J., 29(2):209–216, 1977 . This precise result of Odlyzko was used by Takeuchi in his paper Arithmetic Fuchsian groups with signature (1; e). J. Math. Soc. Japan, 35(3):381– 407, 1983. to prove finiteness results for arithmetic groups. | |
| Oct 6, 2016 at 5:14 | history | asked | Pablo | CC BY-SA 3.0 |