Timeline for Greatest common divisor in $\mathbb{F}_p[T]$ with powers of linear polynomials
Current License: CC BY-SA 4.0
15 events
| when toggle format | what | by | license | comment | |
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| Feb 21, 2020 at 8:10 | vote | accept | Martin Brandenburg | ||
| Feb 19, 2020 at 19:58 | comment | added | Will Sawin | @MartinBrandenburg I think I figured out how to abstractly show the existence of a counterexample in this case - see my answer. | |
| Feb 19, 2020 at 19:56 | answer | added | Will Sawin | timeline score: 11 | |
| Feb 19, 2020 at 19:56 | comment | added | Martin Brandenburg | The first example also works with wolframalpha.com, input is PolynomialGcd(x^(7703)-x,(x+1)^(7703)-(x+1),(x+2)^(7703)-(x+2),Modulus->3). But with the second exponent it says, for some reason, "Wolfram|Alpha doesn't understand your query". | |
| Feb 19, 2020 at 19:48 | comment | added | Martin Brandenburg | Ok. The first example works, but the second is too large for GAP, I'm afraid. There is an error message. | |
| Feb 19, 2020 at 19:38 | comment | added | Will Sawin | Can you try examples like $p=3,n=1+(3^{11}−1)/23,1+(3^{23}−1)/47,1+(3^{29}−1)/59$? | |
| Feb 19, 2020 at 19:11 | comment | added | Will Sawin | @MartinBrandenburg Oh sorry, add one to all the numbers that I said. | |
| Feb 19, 2020 at 13:47 | history | edited | Martin Brandenburg |
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| Feb 19, 2020 at 8:35 | history | edited | Martin Brandenburg | CC BY-SA 4.0 |
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| Feb 18, 2020 at 21:03 | comment | added | Martin Brandenburg | Unfortunately $T^p - T = \mathrm{gcd}(f,D^{p-1}(f))$ is not true for the following pairs $(n,p)$ for $n \leq 200$: $(33,5),(73,7),(81,5),(109,7),(113,5),(127,7)$. | |
| Feb 18, 2020 at 20:49 | comment | added | Martin Brandenburg | Thanks Henrik, and nice to hear from you! I will try that. | |
| Feb 18, 2020 at 8:10 | comment | added | HenrikRüping | Here is a wild guess. Let $D$ be the difference operator,i.e. $Df(x)=f(x)-f(x-1)$. Then the upper gcd is also the gcd of $f,Df,D^2f,..,D^{p-1}f$ for $f=T^n-T$. Maybe that gcd is already the gcd of $f$ and $D^{p-1}f$. Could you run a computer experiment to check this ? Then when computing $D^{p-1}$ for $p>2$, we have to derive twice,i.e. the last summand in $T^n-T$does not matter. This could be why the behaviour is different for $p=2$. | |
| Feb 17, 2020 at 23:33 | history | edited | Martin Brandenburg | CC BY-SA 4.0 |
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| Feb 17, 2020 at 23:12 | history | edited | Martin Brandenburg | CC BY-SA 4.0 |
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| Feb 17, 2020 at 23:04 | history | asked | Martin Brandenburg | CC BY-SA 4.0 |