New Mathematical Proof Relied on Astronomy

[The skepTick]

If you're bored with math, skip the following paragraphs and check out this New Scientist article (http://space.newscientist.com/article/dn14064-astronomy-study-proves-mathematics-theorem.html). I also did a little writeup (http://wayofthewoo.blogspot.com/2008/06/astronomy-conjecture-proves.html) at my blog to fill in some of the article's holes. The upshot is that the work of an astronomer/astrophysicist (I haven't figured out which) was used to prove a math conjecture, turning the little conjecture into a full fledged theorem. Cool...for some.

A few years ago, a couple mathematicians had been working on a proof of a problem stemming from the Fundamental Theorem of Algebra . Specifically, they were looking at the number of solutions possible when dividing one polynomial by another polynomial. These functions are called [I]rational harmonic functions. Even more specifically, they were looking at the case when the two polynomials were relatively prime. What they were able to show for this case was that there could be as many a 5n-5 possible solutions, but no more. However, they couldn't prove that 5n-5 was the exact upper bound...it could have been 4n-4 or 3n-2. All they could show was that you could have no more than 5n-5.

Meanwhile, an scientist studying gravitational lensing happened to solve the exact same problem and demonstrated that there were 5n-5 solutions to her equation. So, in essence, she proved the case for 5n-5, which meant that the mathematicians could use her work and firmly establish 5n-5 as the absolute upper bound.

[DeiRenDopa]

... snip ...

the work of an astronomer/astrophysicist (I haven't figured out which)

... snip ...YMMV but ...

that's definitely the work of an astrophysicist!

An astronomer would have developed a plan for observations (using some combo of telescope(s) and instrument(s)), taken them, reduced the data, drawn some conclusions, and so on.

OTOH, one's job title may not be a particularly reliable indicator of one's work, and in this case our heroine has gone on to greener pastures ...

[andyandy]

Very interesting stuff!

I started a thread a couple of weeks ago on how mathematics often pre-empted the discovery of its application in the real world, and this is a case where the exact opposite is true....

all hail mathematics, the true god of the universe :D

[sphenisc]

I'm confused by your write-up referring to the result of dividing one polynomial by another as "the resulting polynomial being of degree n". If the result is a polynomial of degree n then surely it must have n roots, not upto 5n-5?

[The skepTick]

I'm confused by your write-up referring to the result of dividing one polynomial by another as "the resulting polynomial being of degree n". If the result is a polynomial of degree n then surely it must have n roots, not upto 5n-5?

Thanks - that was confusing, wasn't it? Good catch. I should have said that the degree n is the maximum degree of the polynomials p and q. I.e. n = max{degree p, degree q} where p and q are the two relatively prime polynomials and the resultant polynomial is r = p/q.

So, the degree of r has been proven to have an upper bound = 5n-5.

I'll update my post.

[ddt]

Thanks - that was confusing, wasn't it? Good catch. I should have said that the degree n is the maximum degree of the polynomials p and q. I.e. n = max{degree p, degree q} where p and q are the two relatively prime polynomials and the resultant polynomial is r = p/q.

So, the degree of r has been proven to have an upper bound = 5n-5.

I'll update my post.

Thanks for the write-up. A thing that confuses me: when it would be about a polynomial r(z) = p(z)/q(z), I'd guess that r has deg(p) zeros and deg(q) poles, wouldn't it?

The PDF you linked to speaks of a function r(z) = p(z)/q(z) - conj(z), but I haven't gotten past page 1 yet :(

[balrog666]

If you're bored with math, skip the following paragraphs and check out this New Scientist article (http://space.newscientist.com/article/dn14064-astronomy-study-proves-mathematics-theorem.html). I also did a little writeup (http://wayofthewoo.blogspot.com/2008/06/astronomy-conjecture-proves.html) at my blog to fill in some of the article's holes. The upshot is that the work of an astronomer/astrophysicist (I haven't figured out which) was used to prove a math conjecture, turning the little conjecture into a full fledged theorem. Cool...for some.

A few years ago, a couple mathematicians had been working on a proof of a problem stemming from the Fundamental Theorem of Algebra . Specifically, they were looking at the number of solutions possible when dividing one polynomial by another polynomial. These functions are called [I]rational harmonic functions. Even more specifically, they were looking at the case when the two polynomials were relatively prime. What they were able to show for this case was that there could be as many a 5n-5 possible solutions, but no more. However, they couldn't prove that 5n-5 was the exact upper bound...it could have been 4n-4 or 3n-2. All they could show was that you could have no more than 5n-5.

Meanwhile, an scientist studying gravitational lensing happened to solve the exact same problem and demonstrated that there were 5n-5 solutions to her equation. So, in essence, she proved the case for 5n-5, which meant that the mathematicians could use her work and firmly establish 5n-5 as the absolute upper bound.


But, but, but ... it's only a specific case! It doesn't mean squat!!

Mathematicians everywhere go, "Waaaaaaaaaah!"

[The skepTick]

Thanks for the write-up. A thing that confuses me: when it would be about a polynomial r(z) = p(z)/q(z), I'd guess that r has deg(p) zeros and deg(q) poles, wouldn't it?

The PDF you linked to speaks of a function r(z) = p(z)/q(z) - conj(z), but I haven't gotten past page 1 yet :(

That's correct. It only works for deg(p) or deg(q) > 1. The solutions lie within the complex plane, so I wonder for what circumstances the solutions must be real. Maybe I should read further in the paper, but it sounds like the astrophysical proof means that 5n - 5 potential images can be seen...which would indicate real solutions.

[Lucky]

Not that rare for advances in pure maths to be made by applied mathematicians and theoretical physicists – more unusual, I guess, for an experimental physicist (if that's what she was). It seems she couldn't prove her conjecture, and didn't realise she was making a contribution to the theory of equations.

More info here (http://www.sciencenews.org/view/generic/id/33082/title/Accidental_astrophysicists).


Something very neat (and imo a bit disturbing) - mathematical proof using the laws of physics is possible.

Here's a pretty example (from Martin Gardner):

An irregular polyhedron may be unstable on some of its faces. Is it possible to construct an irregular convex polyhedron that's unstable on every face?

Answer 1: No, {forbiddingly difficult proof by geometry/algebra}.

Answer 2: No, because it would enable a perpetual motion machine.

(eta: didn't mean to imply the example in The skepTick's OP is a 'proof by physics'.)