[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.

[ColinWright]

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

[thaumasiotes]

> 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?

[jlev1]

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.

[im3w1l]

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.

[soVeryTired]

That sounds extremely powerful. How general is the method? Could it provide a handle on the Birch and Swinnerton-Dyer conjecture?