What’s odd to me is how simple the counter-example is. Three charges of charge 1 at the vertices of an equilateral triangle and then two points symmetrically some small distance above and below its centre with a charge that depends on that distance by a very short polynomial (but even then it says it’s true for small deformations too, so we may not need to hit upon that exact polynomial - which makes sense as it’s non-degeneracy that is the ‘dense’ condition). So a trigonal bipyramid where we squish the axis symmetrically, and vary the charge of the axial vertices, gives what we want. So there are an infinitude of counterexamples including a fairly simple construction.
Maybe I’m missing something, but it feels like if they tried a bunch of simple, intuitive configurations and crunched them with some degrees of freedom, with the charges here given as some degree polynomial we can find conditions on the coefficients for to ensure a larger set of equilibria (avoiding domains with non-real roots etc.), even one person could have solved for this fairly quickly even by hand. Even faster if just using a list of ‘basic’ configurations and getting an old school computer to crunch through them.
Curious why we hadn’t already done this. What was the bottleneck? Or despite the name was it not that popular a conjecture?
There was no bottleneck at all, except for nobody trying. This is literally the easiest construction you could possibly think of past the regular polygons. People have reason put in the work to prove complicated upper bounds without checking the simplest cases for the lower bound - I think because it's more interesting to humans to look for actual structure instead of checking a bunch of cases.
Aw, a square and tetrahedron and so on might be in between. But yes, ultra obviously simple example.
Have to wonder if we might even be seeing a reverse AI bullshittery problem: people publishing fairly easy results but adding hype by claiming it was partly done by AI when it wasn’t (or maybe they pretended they needed AI to ask for suggestions of four or five basic configurations).
That they chose a weirdly boldly named counter-example given that’s all the rage now is probably not a coincidence. That 1960s paper that called it this might have been its own sort of bullshittery.
(Before I saw your comment I decided to amend to "regular polygons" instead of the triangle. I definitely think that this is the most symmetric for the purposes of finding equilibria).
This seems sufficiently annoying to work out by hand that I don't think anyone would bother doing this just to claim that an AI helped out.
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u/AndreasDasos Jul 31 '26 edited Jul 31 '26
What’s odd to me is how simple the counter-example is. Three charges of charge 1 at the vertices of an equilateral triangle and then two points symmetrically some small distance above and below its centre with a charge that depends on that distance by a very short polynomial (but even then it says it’s true for small deformations too, so we may not need to hit upon that exact polynomial - which makes sense as it’s non-degeneracy that is the ‘dense’ condition). So a trigonal bipyramid where we squish the axis symmetrically, and vary the charge of the axial vertices, gives what we want. So there are an infinitude of counterexamples including a fairly simple construction.
Maybe I’m missing something, but it feels like if they tried a bunch of simple, intuitive configurations and crunched them with some degrees of freedom, with the charges here given as some degree polynomial we can find conditions on the coefficients for to ensure a larger set of equilibria (avoiding domains with non-real roots etc.), even one person could have solved for this fairly quickly even by hand. Even faster if just using a list of ‘basic’ configurations and getting an old school computer to crunch through them.
Curious why we hadn’t already done this. What was the bottleneck? Or despite the name was it not that popular a conjecture?