Timeline for What's the number of solutions of the quadratic equation $x_1^2+\dots+x_m^2=0$ over finite ring $\mathbb{Z}/p^n$?
Current License: CC BY-SA 4.0
14 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Jun 29, 2021 at 6:58 | vote | accept | CommunityBot | ||
| Jun 27, 2021 at 22:55 | answer | added | Joe Silverman | timeline score: 5 | |
| Jun 27, 2021 at 20:51 | answer | added | Fedor Petrov | timeline score: 4 | |
| Jun 27, 2021 at 18:31 | comment | added | Terry Tao | The question is already answered, but I would like to mention that the answer to this question could also be computed by the circle method, as the Gauss sums $S(a) = \sum_{x \in {\bf Z}/p^n {\bf Z}} e( ax^2 / p^n )$ can be evaluated exactly in a relatively straightforward fashion, and the sum in question is simply $p^{-n} \sum_{a \in {\bf Z}/p^n {\bf Z}} S(a)^m$. | |
| Jun 27, 2021 at 11:27 | answer | added | YCor | timeline score: 6 | |
| Jun 27, 2021 at 8:56 | comment | added | YCor | I will post details later today as an answer. | |
| Jun 27, 2021 at 6:02 | comment | added | user178596 | @WillJagy: Thx for the ref. Actually Ireland&Rosen's book already provides a general method for an analogue of my question over finite fields, cf. Section 10.3-theorem 2 in p.147. But I found that my question over a finite ring need some different tricks unlike cases of finite fields. Maybe it just involves the lifting problem as YCor commented. | |
| Jun 27, 2021 at 5:48 | comment | added | user178596 | @YCor: I have some trouble in explictly calculating the number of listings. I tried the tools of Teichmüller character (also called Teichmüller lift) stating that $(\mathbb{Z}/p^n)^\times\cong \mathbb{F}_p^\times\times\mathbb{pZ}/p^n$ in a canonical way. And the reduction from solutions $\mod p^{n+1}$ to those $\mod p^b$ which is given by adding multiples of $p$ is a $p$-to-1 correspondence. But is this reduction a surjection? The Teichmüller lift doesn't seem to help. | |
| Jun 26, 2021 at 20:53 | comment | added | Will Jagy | from 2015, answer by me, math.stackexchange.com/questions/1556792/… | |
| Jun 26, 2021 at 20:49 | comment | added | Will Jagy | Charles Small does many counting problems in Arithmetic of Finite Fields, 1991 | |
| Jun 26, 2021 at 14:08 | comment | added | YCor | OK, so it should be a matter of lifting solutions mod $p^n$ to mod $p^{n+1}$ (and probably should multiply by $p^{m-1}$, not $p^m$) | |
| Jun 26, 2021 at 13:22 | comment | added | user178596 | @YCor: Here is a different question which coincides with mine when $n=1$ answer. That answer provides GTM84 as the ref-ex.19 in p.106. | |
| Jun 26, 2021 at 12:54 | comment | added | YCor | A direct approach should make the job, rather than a magic formal recipe? Apart from a few exceptions passing from $p^n$ to $p^{n+1}$ should multiply the number of solutions by $p^m$, and the main case should be $n=1$, which shouldn't be hard. Do you know the answer when $n=1$? | |
| Jun 26, 2021 at 12:49 | history | asked | user178596 | CC BY-SA 4.0 |