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
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~
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
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 |
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.
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.