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

The fish

[leonmergen]

http://babelfish.altavista.com/babelfish/trurl_pag econtent?url=http%3A%2F%2Fwww.fontys.nl%2Fnieuws%2 Fnieuws_artikel.asp%3Fdocid%3D3487&lp=nl_en

Re:Damn, 11 Years too late

[Alien54]

Unfortunately, the solution requires a knowledge of calculus:

To do so, we express x as a powerseries of s, and calculate the first n-1 coefficients. We turn the polynomial equation into a differential equation that has the roots as solutions. Then we express the powerseries' coefficients in the first n-1 coefficients. Then the variable s is set to a0. A free parameter is added to make the series convergent.

The short paper has more details.

Mirror

[avalys]

Apparently some people can't get to the site, which is funny because I'm having no problem, but here is a mirror.

The Roots of any Polynomial Equation

Its no big deal

[moss1956]

The theorem of Abel (or Galois) that is being referred to merely claims that you can't find a general formula built from just the arithmetic operations plus taking nth roots. It has been known for a long time that there is a general formula using elliptic functions.

The student just used the method of formal power series to solve the equation. This approach dates back at least to Cauchy ~1850 and probably can be found in the works of Euler.

Re:run away!

[slavemowgli]

That's dutch, not german.

I did something similar

[Ann Coulter]

I used hypergeometric functions to solve the equation

a x^b + c x^d + e x^f = 0

where the exponents are integers and the coefficients can be complex. I tried to generalize it for complex exponents but I quit after a while. Google should provide some preliminary information on using hypergeometric functions to solve the quintic

a x^5 + b x^c + e = 0

where c is less than 5 and greater than zero.

This is an analytic solution to the general trinomial that I found empirically (without proof). If one wants to solve to solve the quartnomial then two dimensional structures, quintnomials need 3 dimensional structures. This was computationally taxing on me and my computer so I didn't even consider the quartnomial equations.

By the way, I have implemented a Jenkins-Traub algorithm not so long ago that gives numerical approximations to general polynomial roots. It is fast and well known.

See Mathematica Poster: "Solving the Quintic"

[high na]

In this poster, they discuss this topic precisely, including Abel's theorem. One of the readers was correct: although Abel proved impossibility of solution for polynomials higher than degree 5 IN TERMS OF ROOTS AND OTHER ALGEBRAIC ENTITIES, there is nothing ruling out a solution in terms of, say, hypergeometrics. This is precisely what they do, and there's a nice development of this using power series. So, although I didn't get to read the PDF, it seems from the posts here that this is what the student did. Thus, no big deal. That said, I salute the student for figuring this out on his own, and he shouldn't be discouraged by discovering something that is not new.

Looks flawed

[hanwen]

Looks flawed to me. He performs a sensitivity analysis in the constant of the polynomial (which he calls "s"). It remains unclear why. After a convoluted sequence of operations, he derives a power series for x as function of s , and proves convergence by requiring |s| smaller than 1.

Finally, he puts back a_0 into s, but conveniently forgets the case that a_0 is bigger than 1.

Also, it is not clear whether this is in the complex plane or not. For example, for finding real roots of real polynomials, you could use Sturm Sequences. There's even sample code in graphics Gems IV (IIRC).

In any case, the student was studying at the "hogeschool" which roughly translates to "higher professional education", a school which doesn't teach mathematics, and whose level which significantly lower than Dutch the MSc., BSc. or engineering degree.

Han-Wen

(yes, I am a mathematician)

Re:Looks flawed

[Anonymous Coward]

Finally, he puts back a_0 into s, but conveniently forgets the case that a_0 is bigger than 1.

Except he doesn't forget this. He tells you to divide the polynomial by a constant to make it so. Since you're a "mathematician" let me help you visualize this difficult step:

18x^6 - 12x^3 + 3 = 0 ... divide by 6

3x^6 - 2x^3 + 1/2 = 0

Gee, a_0 is less than 1 now and the roots are the same. Huh, I wonder how that happened? In actuallity he tells you to divide by a larger number such that all of the coefficients are fractions, but you get the picture.

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

[aardvarkjoe]

I believe it means that there will never be a magic equation to 4th order polynomials and above like one learned in algebra to the 2nd order polynomial.

Fifth order, I believe; there actually is a quartic equation to solving a 4th-order polynomial algebraically. (Although admittedly, even with the equation I'd hate to try to do it by hand.)

Mathematical Elucidation

[logicnazi]

Alright, whoever wrote the article seems very confused about mathematics and abel's theorem in particular. I'm not actually an algebraist myself but I am in mathematical logician so I can comment a bit about impossibility results.

Abel's theorem merely says you can not solve the general quintic (5th degree) or higher in terms of radicals. That is entierly in terms of multiplication, addition, and taking nth roots. If we don't put that restriction about radicals the solution is trivial. Let x be such that P(x)=0 is one obvious solution.

Going through this again the write up is *entierly wrong*. It is completly possible to give an exact solution for the general polynomial (I just did in the paragraph above). Furthermore this distinction between exact and numerical solutions which is made so much of by our high school and college teachers is really illusionary. Writing a solution in terms of sin(3) isn't an exact value, we just have a good algorithm to approximate sin. Really what we mean when we talk about exact solutions is solvable in elementary functions, which is nothing but a certain commonly used set of functions for which we have good approximations. Unfortunatly, we still insist on students 'solving' differntial equations rather than just finding some quickly converging numerical solution even though at a deep level these are not differnt.

Now since abel's theorem there has been considerable research on other ways to solve polynomial equations. For instance one big result was that a certain degree of polynomials could be solved in a terms of continous two place function. Possibly this result in question is another result like this one but I imagine it is much less significant. For one I'm not entierly convinced he is correct, nor novel. (Don't make the mistake of assuming if he is right he has given a continous solution of any polynomial..it isn't clear his solution is continous in the coefficents).