Timeline for answer to What is the oldest open math problem outside of number theory? by fedja
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| when toggle format | what | by | license | comment | |
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| Jul 18, 2025 at 12:35 | comment | added | James E Hanson | Also something that this tells us is that if there is a counterexample, then there's a counterexample with algebraic parameters. | |
| Jul 18, 2025 at 10:02 | comment | added | James E Hanson | In principle this problem is algorithmically decidable (as a function of $n$ and $d$) using quantifier elimination for RCF, since a definable set in an o-minimal theory is infinite if and only if it has a non-isolated point (which can be expressed in first-order logic in RCF). I have no idea whether this results in a feasible computation even in the $n=3$, $d=3$ case though. | |
| Oct 3, 2024 at 18:10 | comment | added | Per Alexandersson | I still have flashbacks about this problem - Boris Shapiro gave me this problem to work on, first year as an undergraduate student. I did not manage to get far :) | |
| Sep 15, 2024 at 13:10 | comment | added | Alexandre Eremenko | @Mark Lewko: more precisely, they did not prove finiteness, but assuming finiteness, they proved un upper bound. As far as I know, the problem is unsolved even for 3 charges. I've a computer assisted for 3 equal charges, but did not verify that it is correct, and it is not published. | |
| Sep 14, 2024 at 16:58 | comment | added | Mark Lewko | It seems (from the paper of Gabrielov, Novikov and Shapiro: arxiv.org/pdf/math-ph/0409009) that Maxwell asserted without proof an upper bound (in fact $(n-1)^2$) on the number of equilibrium points in his treatise, rather than posing it as a question/conjecture. That said, it is a good example particularly given that it is precisely formulated in the original source. | |
| Sep 14, 2024 at 10:11 | history | answered | fedja | CC BY-SA 4.0 |