Slashdot Mirror
General Solution for Polynomial Equations?
An anonymous reader writes "On september 9, several media reported that a young Dutch student found a formula to determine the roots of any polynomial equation. Does this conflict with Abel's proof that such a formula cannot exist? Here is the news item (in Dutch) on his school's homepage." Another reader writes "A Dutch student at the Fontys school of physics has solved a math problem of several centuries old: finding the roots of any polynomial equation. Arxciv copy here. Although an exact solution has been proven impossible for higher orders, this is not the case for numeric solutions."
Popular media today reports that someone has done what is well established to be impossible. Now, which one is more likely:
i) Abel's proof contains a flaw that generations of extremely talented mathematicians have failed to spot in their years and years of teaching it.
ii) Student mistaken; popular media talking out of arse.
(Can't read PDF; slashdotted)
Athletic Scholarships to universities make as much sense as academic scholarships to sports teams.
I'm surprised he didn't include some sample Matlab, Java applet or C code in his paper. It would be useful to have a demonstration that this really works.
Vintage computer adverts: http://www.vintageadbrowser.com/computers-and-software-ads
Think about this
You can do more in 5 minutes mathematically with t Pentium 4 than a mathmatitician 200 years ago could do in a lifetime
You got it... instead of a solution by radicals (which Abel's proof shows does not exist for general polynomials with degree 5 and higher) he takes it into differential equations and creates a powerseries, which essentially gives an approach to the real number root, which doesn't necessarily have a radical decomposition. Plus, the proof looks like a lot of handwaving at a cursory glance. I'm more inclined to believe that this is a wash.
It's a NUMERIC solution, not an ALGEBRAIC solution.
:).
:)
Abel's proof showed that polynomials with a degree higher than 4 could not be solved algebraically (i.e through a finite number of additions, subtractions, multiplications, etc.). Abel's proof did no say it was impossible to solve the equations (indeed, numerical solutions to these equations are solved regularly).
This is similar to how some integral equation solutions cannot be expressed in simple terms. However numerical answers are rather easy to obtain (even easier with a computer)
The method presented is a simpler way to find the roots of polynomial equations numerically by treating it like a power series (x, x^1, x^2,...,x^n) and applying standard differential techniques.
Pretty cool if you ask me.
~X~
~X~
From the looks of the ugly type, he must of used word. Oh the horror. The horror.
nohup rm -rf ~/. >& zen &
Note that the student's result is not a closed formula, and is thus not in conflict with Abel's proof. The system uses convergence (and thus, reuires an infinite number of operations) to find the correct roots.
--- Sigmentation Fault - Comments Dumped
Not so. Exact solutions like the ones provided by mathematical formulas are still useful for a number of reasons:
From a seasoned math professor's reading of it: "It looks like a mess to me.
I don't know what his point is. He says its a "method of solving the roots"
of a polynomial. Well, we already have very fine methods for doing that,
interval Newton methods for instance. Using circular disk arithmetic in the
complex plane we can find all the complex roots as well.
There is no need whatever to make things more complicated such as going to
differential equations. That is unneccessary. Root finding is an algebraic
problem."
stuff |
You can do "better" than that. If you're prepared to write the roots in terms of logical functions, you can "solve" anything.
Want the roots of f(x) = 0?
They are
There are even computer implementations of this for limited cases (called "generate and test" algorithms). But I wouldn't advocate running big headlines claiming -- MarkusQSome Dutch kid is very bright and has found a rapidly converging power series for finding roots. It has been done before but may be this one is slightly more or less cool that the others.
What this does have any sodding effect on is Abel's Impossibility Theorem which is a good thing because it would screw up lots of other things that take Abel as axiomatic.
All credit to the kid I certainly never produced anything a tenth that far about my school grade and this is the slack news season.
Time for some mega nerdiness: I was captain of the math team when I was in high school.
Feh, screw that "nerdiness" crap. Good for you. Math is a powerful tool, worthy of dedication. I wish I were better at it, and respect those who are. I think being captain of the math team is far and away a better thing than being the captain of the freakin football team.
>> Dutch student found a formula to determine the roots of any polynomial equation. Does this conflict with Abel's proof that such a formula cannot exist?
If something exists, the fact that it can exist is irrefutable.
From what I can tell, it appears to be a method for transforming a polynomial into a differential equation some of whose solutions are roots of the polynomial. From that point I suppose one could use numerical methods for ODE's to find those solutions.
People here have been commenting that Newton's method works just fine for finding roots of polynomials. But, convergence can be quite slow, especially for unidentified multiple roots though, and for highly clustered roots you can run into conditioning problems.
The paper makes no mention of actual numerical algorithms (in particular no discussion of convergence rates or guarantees for solving the ODE numerically) so it is hard to say whether the result is actually useful or just a bunch of manipulation of symbols on paper.
All is Number -Pythagoras.
Hey, the Dutch are dudes.
Ydco co
Nope, grandparent was right. You can't just divide and expect convergence to suddenly occur. I believe the leading coefficient is assumed to be 1 for the result he's using. IAA math major.
--If the world didn't suck, we'd all fall off.
Don't think of it as a flame---it's more like an argument that does 3d6 fire damage
This kid is in what I'm guessing is the equivalent of a really good technical high school, so he deserves praise for just being at the level of messing around with this kind of stuff, let alone attempting to publish a paper. I'm sure some of his paper is lost in translation, but, commenting on the paper itself, it would never get accepted into a reputable journal for publication.
First, he never gives any clear indication whatsoever of what he intends to prove that his method will do. What does it mean to provide a method for "solving the roots" of a general polynomial equation (determine one root, determine all roots, determine all real roots, etc)? Is the polynomial expressed over an arbitrary field, the real numbers, or the complex numbers? What is a "powerseries of s"? The power series of a function is defined as an infinite polynomial that converges over a certain interval to the value of the original function. S, as far as I can tell, is a constant, so thus a power series of s would be s itself.
It might seem like nitpicking, but it's ambiguities like these that make it nearly impossible to read this paper and determine its correctness.
I could pore through it, guessing at the meaning of unorthodox terminology and then proving everything that he hasn't proven to the level of mathematical rigor. But that's his job, and the job of his mentors/advisors and the referee at whatever journal he tries to submit this to.
Furthermore, issuing a press release hyping this as the solution to a long-sought-after open problem is irresponsible. The only problem this has the potential to solve is finding a faster algorithm for computing roots (real, complex, I'm not sure, he didn't say), but the paper does not address whether this is, in fact, any better at all than current root-finding methods. Even if so, it certainly wouldn't make mathematical waves except in some very select circles, were it not that he was in high school.
So, I apologize if this sounds harsh. I certainly couldn't have done this in high school, and it has the potential to shave some overhead off finding the roots of certain polynomials, which would be a nice little result. But I felt it necessary to explain why this paper isn't yet real mathematics.
But that's not the whole story. Of course, if sin(1) is a root of your polynomial, then most people would be happy and consider that a perfectly good number that we can sink our teeth into. It is in fact possible that there exist polynomials of degree 5 whose roots cannot be expressed using buttons on your calculator (assuming your calculator is somehow infinite precision). That means using exponentiation, natural logs, arcsinh, etc. A particular example is the polynomial
2x^5 - 10x + 5
Well, with current math we can't prove that the roots of this polynomial have this property. But if you assume the as-yet-unproven (or as-yet-disproven, take your pick) "Schanuel's Conjecture" from number theory, you can indeed prove that this polynomial's roots are in some sense "inexpressible".
Yeah, yeah, of course you can approximate them numerically. Any australopithecine could realize that in the time it takes to gnaw an antelope femur down to the marrow.
(Personally speaking, I find the possibility that we can't explicitly write down the roots of a quintic polynomial -- especially such a nice one -- somewhat disturbing.)
Reference for the claims above: Timothy Chow, What is a closed-form number?, American Mathematical Monthly, May 1999, vol. 106, pages 440-448.