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."

Comments

Man does the impossible

[gowen]

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)

Re:Man does the impossible

[Chris_Jefferson]

Actually, I'm fairly certain it is:
iii) Student comes up with interesting (and possibly new, I don't know) result of generating infinate series which converges to root of polynomial. Someone (popular media?) believes that this violates Abel's proof (it doesn't, his proof is for finite representations of the roots).

This has NOTHING to do with Abel, or Godel, or anyone other related theories, as they do not consider the case of infiniate series.

Re:Man does the impossible

[WolfWithoutAClause]

Wow. That's news. At least to anyone completely unaware of Newton's method.

Yeah, but Newton's method isn't guaranteed to converge. This method claims to converge; although you don't get the exact answer in a finite number of steps.

Whether this method is useful or not, probably depends on how fast it converges and how long it takes to do each step.

Re:Isn't it an approximation method?

[Leadhyena]

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.

Re:Right in the middle of my Calc class too...

[Xyrus]

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~