Mathematicians Crack the Cursed Curve(quantamagazine.org)
quantamagazine.org
Mathematicians Crack the Cursed Curve
https://www.quantamagazine.org/mathematicians-crack-the-cursed-curve-20171207/
37 comments
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?
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?
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.
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.
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.
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.
A modern reason for being interested, wildly over-simplifying[0] ...
Consider an equation of the form y^2=ax^3+bx+c, and consider the points (x,y) where x and y are rational. There may be none, there may be finitely many, there may be infinitely many.
Take a huge, structureless[1] prime p. Any rational r/s can be thought of as r times s^{-1} modulo p, so rationals are roughly the same as integers when you work modulo a prime.
So the rational solutions to the equation above (which, by the way, is an elliptic curve) give us integer solutions when we work modulo p.
And now by using the geometry of the curve we get a group where the elements are pairs of integers. That's because we found rational solutions. Suddenly everywhere we use groups - such as in cryptography - we can use these numbers that have arisen as rational solutions to an equation.
So being able to find rational solutions to equations is useful.
[0] With any luck someone more knowledgeable can fix the worst of the errors in this.
[1] So not of any particular form, such as 3^k+1 or similar
Consider an equation of the form y^2=ax^3+bx+c, and consider the points (x,y) where x and y are rational. There may be none, there may be finitely many, there may be infinitely many.
Take a huge, structureless[1] prime p. Any rational r/s can be thought of as r times s^{-1} modulo p, so rationals are roughly the same as integers when you work modulo a prime.
So the rational solutions to the equation above (which, by the way, is an elliptic curve) give us integer solutions when we work modulo p.
And now by using the geometry of the curve we get a group where the elements are pairs of integers. That's because we found rational solutions. Suddenly everywhere we use groups - such as in cryptography - we can use these numbers that have arisen as rational solutions to an equation.
So being able to find rational solutions to equations is useful.
[0] With any luck someone more knowledgeable can fix the worst of the errors in this.
[1] So not of any particular form, such as 3^k+1 or similar
> 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!)
Ok, any finite set of points can be interpolated by a curve.
Why is it obvious that one of these interpolated curves will necessarily avoid intersecting C at any other point?
Ok, any finite set of points can be interpolated by a curve.
Why is it obvious that one of these interpolated curves will necessarily avoid intersecting C at any other point?
Good point, it's not necessarily possible! That said, it would be enough to know that the intersection just contains all the rational points (and is finite, though that's automatic for 2+ polynomials in two variables with no common factors). Then we can just check them one by one, discarding the non-rational points.
Alternately, it's possible the construction gives a system of auxiliary equations, which, together with f(x,y) = 0, pick out the rational points of the curve. (The term "variety", as in "Selmer variety", means solution set to a system of polynomial equations). Still, short of knowing the points in advance, I wouldn't know how to easily produce such equations.
Alternately, it's possible the construction gives a system of auxiliary equations, which, together with f(x,y) = 0, pick out the rational points of the curve. (The term "variety", as in "Selmer variety", means solution set to a system of polynomial equations). Still, short of knowing the points in advance, I wouldn't know how to easily produce such equations.
I can think of counterexamples if we allow C to self-intersect: We can then make a loop and pick one point outside it and one point inside and zero on it.
That sounds extremely powerful. How general is the method? Could it provide a handle on the Birch and Swinnerton-Dyer conjecture?
> 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".
What does this even mean? This looks like fancy words for "Kim used lines in his solution".
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.
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.
Maybe its referring to the way Feynman describes light travelling, for example in this lecture.
https://www.youtube.com/watch?v=kMSgE62S6oo
It's something like, (and please don't try too hard to understand my description here, I watched the video over a year ago, and you've got Feynman at the link) imagine a photon is a particle going in all directions, and has an arrow inside it spinning at a rate of 10^15 revolutions per second, and the probability that light goes from its origin to a particular destination is the amplitude of the sum of all the internal arrows of the all the possible paths between the origin and the destination.
Even if this is not what the article is referring to, its a reminder that for a lot of people, light doesn't move in straight lines.
Finally, the series linked series of lectures are amazing, if you have a layperson's interest in Physics then this will take your knowledge to a new level.
https://www.youtube.com/watch?v=kMSgE62S6oo
It's something like, (and please don't try too hard to understand my description here, I watched the video over a year ago, and you've got Feynman at the link) imagine a photon is a particle going in all directions, and has an arrow inside it spinning at a rate of 10^15 revolutions per second, and the probability that light goes from its origin to a particular destination is the amplitude of the sum of all the internal arrows of the all the possible paths between the origin and the destination.
Even if this is not what the article is referring to, its a reminder that for a lot of people, light doesn't move in straight lines.
Finally, the series linked series of lectures are amazing, if you have a layperson's interest in Physics then this will take your knowledge to a new level.
This is just another guess, but it might refer to something of this nature:
http://www.analyzemath.com/calculus/Problems/minimum_distanc...
Maybe it means something along the lines of "instead of thinking of those equations as curves in a flat space, he thinks of them as lines in a curved space". Disclaimer, I know nothing about number theory.
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.
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.
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.
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.
It's also got 3 dimensions. That's going to make it somewhat hard to plot on a 2-dimensional display.
Erm, not really. You've heard of 3D graphics I assume?
It’s akin to saying the map of the earth is hard to print on paper. You’d need to put the map on a sphere (so, a globe) to reliably represent the graph.
Erm, unless this is in some way not a 3D curve, it is trivial to plot it in 3D and show a 2D projection of it on a screen. I don't know how you could disagree with that.
Edit: Apparently it is a surface. Someone posted a link below https://www.desmos.com/calculator/4qu7gezqmx
Still not exactly difficult to plot - you probably need some transparency to be able to see it though.
Edit: Apparently it is a surface. Someone posted a link below https://www.desmos.com/calculator/4qu7gezqmx
Still not exactly difficult to plot - you probably need some transparency to be able to see it though.
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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...
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...
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...
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...
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?
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.
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This article is about (mathematical) application. They applied Kim's algorithm to find all rational solutions to a particularly hairy equation.
Kim's solution appears to be a better method to use for discovering the roots of a Diophantine equation that has finite roots.
This could have impacts on computational encryption....it could also just make mathematics in this area easier to work through.
This could have impacts on computational encryption....it could also just make mathematics in this area easier to work through.
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).
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.
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).