Timeline for Reasons to prefer one large prime over another to approximate characteristic zero
Current License: CC BY-SA 4.0
12 events
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| Apr 15, 2021 at 21:01 | comment | added | KConrad | @AntonMellit you say it is not surprising that two irreducibles in $\mathbf Q[x]$ have a common root modulo some prime. Two relatively prime polynomials $f(x)$ and $g(x)$ in $\mathbf Q[x]$ are relatively prime mod $p$ for all but finitely many $p$, so the probabilistic heuristic you describe is modeling an event that can happen for each $f$ and $g$ only finitely many times. Thus it is not surprising that some polynomials fail the heuristic: $f(x) = x^3 - x + 1/3$ and $f'(x) = 3x^2 - 1$ are relatively prime and have resultant 1 (or $-1$), so they have no common root mod $p$ for all $p$. | |
| Apr 15, 2021 at 19:49 | comment | added | Anton Mellit | It's not so surprising if you use the right heuristic. An irreducible polynomial with integer coefficients has (on average) $1$ root modulo $p$ by Chebotarev density theorem. Now if you have two polynomials $P, Q$ you can take that one root of $P$ and plug it into $Q$. It will be zero with probability $1/p$. What's the probability that it is not zero for all primes $p$? It's the product of $1-1/p$, but the series $\sum_p 1/p$ is divergent hence the product is zero. | |
| Apr 14, 2021 at 21:01 | history | edited | KConrad | CC BY-SA 4.0 |
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| May 5, 2014 at 1:38 | history | edited | Lucia | CC BY-SA 3.0 |
One last time; I checked this on Meta Sandbox. Hopefully this works out.
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| May 5, 2014 at 1:23 | history | edited | Lucia | CC BY-SA 3.0 |
The earlier version looks messed up on my screen. This is an attempt to fix the formatting; I hope it works; please revert if it doesn't.
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| May 4, 2014 at 18:38 | history | edited | KConrad | CC BY-SA 3.0 |
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| May 4, 2014 at 17:42 | comment | added | Charles Staats | A useful point that underscores the following comment: approximating characteristic 0 by characteristics like 31667 is useful because exceptions seem to be extremely rare--not because they are impossible. It's also why I'm looking for heuristics rather than theorems. | |
| May 4, 2014 at 16:38 | comment | added | KConrad | I agree it would be a better situation if this phenomenon were more widely known. One reason it might not be is that you can learn the theory of resultants without computing them (which was the case for me when I first saw them). C'est la vie. | |
| May 4, 2014 at 16:33 | history | edited | KConrad | CC BY-SA 3.0 |
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| May 4, 2014 at 16:21 | comment | added | Jacques Carette | This happens all the time when trying to factor multivariate polynomials (and the use Hensell lifting to recover the factors). Such cases are all over the computer algebra literature. It is surprising (to me) that this is not more widely known -- I guess one more example of how narrow the silos in academia are. | |
| May 4, 2014 at 16:03 | history | edited | KConrad | CC BY-SA 3.0 |
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| May 4, 2014 at 15:28 | history | answered | KConrad | CC BY-SA 3.0 |