
Mathematicians Crack the Cursed Curve - CarolineW
https://www.quantamagazine.org/mathematicians-crack-the-cursed-curve-20171207/
======
jlev1
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).

~~~
skosch
Thank you, that was helpful.

Follow-up question: is there any practical significance of rational solutions?
I can understand why one might be looking for integer solutions to an
equation. Can you provide an example where rational solutions correspond to
something interesting in the context modeled by the equation – for example the
"path travelled by light" thing hinted at in the article?

~~~
jlev1
Hmm. I don't know about this particular equation (which sounds like it's
mainly significant because it's viewed as a bellwether -- if the method works
on it, it's likely to work on other problems). Anyway.

First -- for "homogeneous" equations like the one being studied (or simpler
ones like x^2 + y^2 = z^2), a rational solution can be rescaled to get an
integer solution -- replace (x,y,z) by (cx,cy,cz), a new solution with
denominators cleared out. Homogeneous equations are very, very common.

That said, yes, the ultimate goal is to understand integer solutions (and as
you say, they're often the only meaningful solutions in practical situations).
But integer solutions can be _impossibly_ hard to find, whereas rational
solutions are just... _very_ hard.

I guess I could imagine some unusual situation where rational solutions make
sense but real ones don't. But it would have to be some context where x,y are
"sort of discrete", they can be broken down into finitely-many parts (so
fractions make sense) but no further (so sqrt(2) is out). But this does seem
less likely.

~~~
leereeves
Does this particular method only find rational solutions less than one?

It seems to me that integer solutions are rational solutions, and if you can
find a finite number of rational solutions and prove those are all the
rational solutions, you've also found all integer solutions (by filtering the
rational solutions for integers).

But when there are infinitely many rational solutions, that may leave an open
question whether there are also infinitely many integer solutions.

------
saagarjha
> 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".

~~~
Sniffnoy
The earlier article that the article links to says a bit more about this. It's
still pretty far from concrete, but it does say a bit more. When they talk
about the path that light travels between two points, they're talking about
the principle of least action.

Going by the earlier article, the idea seems to be roughly like the following.
You associate to each point the fundamental group based on that point. All
these fundamental groups then live in some larger space, and the ones based at
rational points will minimize some quantity analogous to action (or, if we're
thinking of light, time).

That's basically the best I can figure just based on the earlier article.

------
enriquto
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.

~~~
_delirium
It's not straightforward to get a useful visual plot, but the article does
include a somewhat artistic one (w/o labeled axes) in the banner image at the
top. It's a polynomial in 3 variables, so plotting the polynomial itself is
4-dimensional, and the roots are in 3 dimensions. The visualization at the top
of the article is a 3d plot of the locations of the roots.

The seven rational solutions are given on p. 30 of the paper, and are: (x,y,z)
= (1,1,1), (1,1,2), (0,0,1), (-3,3,2), (1,1,0), (0,2,1), (-1,1,0). Those were
already known though. The new result of the paper is to prove that there
aren't any others.

~~~
enriquto
It is a curve in homogeneous coordinates. You can set z=1 to obtain an affine
view of it, and draw it as a regular curve in the plane. Points with z=0 are
points at infinity (directions). The two rational points with z=0 that you
describe mean that the curve has two asymptotes at 45 and -45 degrees.

------
mraison
If you're curious about what the actual paper looks like, here it is:
[https://arxiv.org/abs/1711.05846](https://arxiv.org/abs/1711.05846)

------
johnhenry
For anyone who want's to play around with the curve a bit:
[https://www.desmos.com/calculator/4qu7gezqmx](https://www.desmos.com/calculator/4qu7gezqmx)

------
kungito
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...](https://www.quantamagazine.org/secret-link-uncovered-between-pure-
math-and-physics-20171201/)

~~~
rocqua
The benefits are that mathematicians think it is interesting, and this allows
us to answer mathematical questions we could not answer before. There might be
practical applications I do not know of, but that's not the point of
fundamental mathematical research.

For a thorough defense of mathematics, consider 'A mathematicians apology' by
G.H. Hardy or [1], a modern response to that. To summarize [1], the argument
is that the main product of mathematics is an environment that creates
mathematicians. Those mathematicians can the use their problem solving skills
in practical applications.

[1] [https://ldtopology.wordpress.com/2017/03/18/a-new-
mathematic...](https://ldtopology.wordpress.com/2017/03/18/a-new-
mathematicians-apology/)

~~~
saagarjha
That’s a defense of math in general; do you know what the benefits of this
particular proof are? I’m not looking for practical applications; rather,
could it help solve other problems in the field? Is it an important result in
its own right?

~~~
rocqua
This shows that an oft talked about new technique in Diophantine equations
actually works. The excitement is more due to proof-of-concept than the
immediate logical implications.

------
qubex
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).

~~~
saagarjha
There’s a real dearth of summaries for papers that are aimed at those who have
taken a reasonable amount of math: say calculus, some number theory, linear
algebra. It always either ends up being some sort of terrible analogy to a
real-world phenomenon that has no math in it or something that’s about as
complicated as the paper itself.

