> Drawing inspiration from physics, he thinks of rational solutions to equations as being somehow the same as the path that light travels between two points.
What does this even mean? This looks like fancy words for "Kim used lines in his solution".
It would be nice if the had drawn the damn curve (and the famous 7 points on it). Not all of us are capable of plotting a fourth degree curve on our heads.
If you're curious about what the actual paper looks like, here it is: https://arxiv.org/abs/1711.05846
For anyone who want's to play around with the curve a bit: https://www.desmos.com/calculator/4qu7gezqmx
Could someone please point out the benefits and implications of this? Does this bring us closer to some very important solution or does it have some real world applications?
Edit: I suppose this is a related article talking about the same problem from December 2017. What changed since then?
https://www.quantamagazine.org/secret-link-uncovered-between...
I have an applied mathematics background and I find it quite amusing that the descriptions of this article are so opaque and inscrutable that the ten-plus comments here are all penned by people that are absolutely fixed by what concept the article is trying to express (including myself, incidentally, even after checking the original paper because it’s a totally different domain and level of sophistication compared to my zone).
This is an amazing paper
This is an amazing paper
Just to add a comment on "why this idea is cool" from my perspective (I'm a mathematician).
The situation being studied is: C is a curve in the plane (as another commenter pointed out, the z variable can essentially be ignored and set to z=1), described by a horrendous equation f(x,y) = 0 with very few rational solutions.
Well, thinking abstractly, if there are only finitely many rational solutions, then there certainly exists a second equation, g(x,y) = 0, giving another curve C' that intersects C at only the rational points. (Because any finite set of points can be interpolated by a curve, e.g. by Newton interpolation. [shrug] Nothing deep about this!)
But, it seems completely hopeless to try to find the equation g(x,y) in practice, other than by first finding all the rational points on C by other means, and then just writing down a different curve passing through them.
So what's special here is that this "Selmer variety" approach provides a method, partly conjectural, for constructing C' directly from C. And the paper being described has successfully applied this method to prove that, at least in this one case, C' intersects C at precisely the rational points. (And once you have the two equations, it's easy to solve for the intersection points -- we now have two equations in two variables).
PS: Part of what's special here is the connection between number theory and geometry. A Diophantine equation has infinitely-many solutions if you allow x and y to be real numbers -- there's the entire curve. It's usually an extremely delicate number theory question to analyze which solutions are rational. But here, we're converting the problem to geometry -- intersecting two curves (much easier).