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
17 events
| when toggle format | what | by | license | comment | |
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| Jun 29, 2021 at 6:58 | vote | accept | CommunityBot | ||
| Jun 28, 2021 at 10:18 | history | edited | YCor | CC BY-SA 4.0 |
added $p=2$ case
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| Jun 27, 2021 at 22:18 | history | edited | YCor | CC BY-SA 4.0 |
fixed typos
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| Jun 27, 2021 at 22:16 | comment | added | YCor | The case mod $2^4$ seems to adapt to the general case (mod $2^n$ for $n\ge 4$), I'll take time later to expand it. | |
| Jun 27, 2021 at 21:00 | history | edited | YCor | CC BY-SA 4.0 |
started p=2 case
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| Jun 27, 2021 at 16:59 | history | edited | YCor | CC BY-SA 4.0 |
added remark on p=2 case
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| Jun 27, 2021 at 16:51 | comment | added | Martin Brandenburg | The following form of Hensel's Lemma should be useful to conceptualize the proof: Let $f \in \mathbb{Z}[X_1,\dotsc,X_m]$, let $V_n$ be the set of solutions of $f=0$ in $\mathbb{Z}/p^n$. There is a canonical map $V_{n+1} \to V_n$. Call $x \in V_1$ singular iff $\partial f / \partial X_i \, (x)=0$ for all $i$, otherwise regular; and $x \in V_n$ is called regular iff its image in $V_1$ is regular. Any regular $x \in V_n$ has $p^{m-1}$ preimages in $V_{n+1}$, and a singular $x \in V_n$ has either $p^m$ or $0$ preimages in $V_{n+1}$, depending on if $x$ is already a solution modulo $p^{n+1}$. | |
| Jun 27, 2021 at 16:43 | comment | added | YCor | @MartinBrandenburg thanks, the number of typos was indecent, and I also messed up the formulas of the second lemma, so the eventual value is modified now. Hopefully fixed. | |
| Jun 27, 2021 at 16:40 | history | edited | YCor | CC BY-SA 4.0 |
fixed typos, added sentence. Fixed Lemma 2.
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| Jun 27, 2021 at 16:35 | history | edited | YCor | CC BY-SA 4.0 |
fixed typos, added sentence. Fixed Lemma 2.
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| Jun 27, 2021 at 16:28 | history | edited | YCor | CC BY-SA 4.0 |
fixed typos, added sentence
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| Jun 27, 2021 at 16:14 | comment | added | Martin Brandenburg | I have corrected some typos in proof 1, but probably some others are still there - you want $p^{m-1}$ lifts perhaps? | |
| Jun 27, 2021 at 16:08 | history | edited | Martin Brandenburg | CC BY-SA 4.0 |
added 9 characters in body
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| Jun 27, 2021 at 15:50 | comment | added | YCor | @WillSawin fixed, thanks | |
| Jun 27, 2021 at 15:49 | history | edited | YCor | CC BY-SA 4.0 |
fixed typo
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| Jun 27, 2021 at 11:52 | comment | added | Will Sawin | What happened to $u_1$ in the last two equations? | |
| Jun 27, 2021 at 11:27 | history | answered | YCor | CC BY-SA 4.0 |