Timeline for Reasons to prefer one large prime over another to approximate characteristic zero
Current License: CC BY-SA 3.0
10 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Apr 15, 2021 at 18:15 | answer | added | Anton Mellit | timeline score: 7 | |
| May 4, 2014 at 15:28 | answer | added | KConrad | timeline score: 41 | |
| May 4, 2014 at 13:44 | answer | added | Jacques Carette | timeline score: 23 | |
| Jun 27, 2013 at 4:10 | comment | added | Charles Staats | @al-Hwarizmi: In principle, those computations are possible. In practice, the sort of question for which this method is useful will have the same answer for $\mathbb F_p$, $\mathbb F_{p^m}$, and $\overline{\mathbb F}_p$, so using $p^m$ instead of $p$ would incur additional computational cost for little or no benefit. | |
| Jun 26, 2013 at 20:32 | comment | added | al-Hwarizmi | Charles - this is an interesting question, really inspiring to think about. I am not in algebraic geometry that deep but interested in wht you wrote; one question is there similarly a way to use prime powers $p^m$ instead of $p$? | |
| May 19, 2013 at 15:32 | comment | added | Charles Staats | Timothy: The sort of question for which this technique is applicable should have the same answer for $k$ as for $\bar{k}$. Thus, if the question you are asking has an answer that changes when a univariate polynomial splits, it's probably the wrong sort of question to begin with. | |
| May 16, 2013 at 18:46 | comment | added | Timothy Chow | One kind of problem that can arise is that your polynomials of interest may include a univariate polynomial that "accidentally" splits modulo $p$. I'd be surprised if there were a uniform choice of $p$ that minimized such accidents; from a practical point of view, if you're worried about such things, it is probably better to repeat the calculation with a different choice of $p$ than to strain too hard to select the One True Value of $p$. | |
| May 16, 2013 at 2:17 | comment | added | Theo Johnson-Freyd | ... go into developing precisely this type of heuristic. | |
| May 16, 2013 at 2:17 | comment | added | Theo Johnson-Freyd | I think this is an interesting question, to which I do not have an answer. I will point out that in some sense no prime is better than any other: for any particular finite set of primes, certainly there are sentences that fail exactly on that set. So your question presupposes something about "interesting" questions that can be answered by an algorithm, or about questions that are "likely" to come up in "research". I doubt that pure model theory and pure number theory can give an absolute answer to things about "interesting" questions and "likely research", but conversely much work does ... | |
| May 16, 2013 at 0:57 | history | asked | Charles Staats | CC BY-SA 3.0 |