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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."
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
- Leon Mergen
http://www.solatis.com
Last quarter's PreCalc class said this was impossible? Now it's possible?
Dang it, that means I'll have to buy a new math book for this quarter's Calc class, won't I?
Ah, the world, she is a changin'...
a formula to determine the roots of any polynomial equation. Does this conflict with Abel's proof that such a formula cannot exist?
Although an exact solution has been proven impossible for higher orders, this is not the case for numeric solutions.
No conflict here. Saying that an exact solution does not exist is consistent with saying that numeric solutions do exist.
A numberic solution is a solution that is "close enough", but not exact. Sort of like saying 2.0000000000000001 = 2. They aren't equal, but for many purposes, they are equivalent.
Without RTFA I can categorically state that it's all Dutch to me...
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 have discovered a truly remarkable formula to solve any polynomial, but my site has too little bandwidth for me to post it here.
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
This space intentionally left blank.
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.
The rule of equations (at least in school) is:
The more complicated the equations for the math problem looks, the more likely the answer is 1.
I am a phycisist, not a professional mathematician, and I didn't understand all steps in the whole paper. However, the author mentions a series expansion with an infinite number of terms in equation (6), although only the first n terms are ever used in defining the solution. That sounds a bit strange to me. In any case, the exact solution for a third-order equation (n=3) involves lots of cube roots and I don't see those anywhere, which also suggests that it's all about an approximation method.
Avantslash: low-bandwidth mobile slashdot.
The present:
:/
european academic finds solution to very hard problem.
2 years later:
a) americans find way of turning said solution into entertainment technology and make billions of dollars.b) European academic still unemployed and eating pasta all week.
We need more GREED in europe..
Will code a sig generator for food
The article in question is slashdotted, but my guess is either that this is media sensationalism, or the writeup is claiming something different from the student -- it seems like perhaps a new way to numerically approximate polynomial roots has been discovered.
... + a_0 where a_n are the elements of said vector. Then, by repeated application of omega and polynomial long division, I have an analytical solution to any polynomial, of any order, in complex space.
However, from what I remember, Abel's theorem was proven using Galois Groups and Field extensions. This implies that what it actually proves is that analytical solutions using a particular set of functions -- in particular, the field operations (addition, subtraction, multiplication, division by non-zero) extended to include radicals (square, cubic, etc roots), composed in any way possible (as in a ruler and compass construction proof) cannot possibly generate an analytical formula depicting the solution for polynomials of order greater than 4.
Does this mean that an analytical formula using other functions is impossible? Not at all. Trivially, I will define a function called, say, omega, which, given a n-dimensional complex vector, gives a solution to one of the roots of the function a_n * x^n + a_(n-1) * x^(n-1) +
Clearly, this solution is analytical in the sense that it a) provides an exact solution and b) is algebraic in nature. However, it isn't useful, because it depends on a function (omega) which cannot itself be defined analytically in terms of other functions (or at least, not ones we know how to compute).
The reason Abel's proof is so important is because it deals with the 4 fundamental operations that polynomials themselves use (the field ops) and adds radicals, which are inverse ops to the building blocks of polynomials themselves. So it essentially says, we cannot use the functions that we constructed the polynomial with to solve it.
Now, my omega function may seem a little bit contrived to non-math types, but actually a large number of functions are arbitrarily defined this way. Logarithms are a good simple example. An analytical formula for the likes of log n wouldn't be possible either, and yet we study logarithms without having an express analytical means of calculating them.
What you should ask yourself is, what does analytical mean, anyway? It really isn't useful (or correct) to say that no analytical solution exists unless you explicitly restrict what particular set of 'basic' functions/operators the analytical solution can contain. In Abel's case (and it's a beautiful proof, by the way) he uses the field operators plus radicals. But what if you added logarithms into the mix? Exponential functions?
It's impossible to say. If you don't restrict your base, you open yourself up to the attack that I just used with the omega function (which certainly exists, after all, I just defined it.)
You seem to have forgotten the final step:
5) Profit!
(awaits an ass-whooping by the mods)
How to solve a polynomial
1) put poly in standard form and take the first n-1 derivatives.
2) put the derivatives in terms of x(s) (for 1..n-1), or remember why you dropped calculus and goto step 9.
3) Use the derivatives to write a differential equation with coefficients m1..mn, or remember why you dropped differential equations and goto step 9.
4) Use the original equation to reduce the differential equation to order n, and note the use of "then" instead of "than" in the mit write-up. (sorry, mit).
5) Substitute a formula for x(s), multiply resulting eq by it's denominator, getting another diffEq. Whee! ask a Grad student.
6) Now substitute a power series representation. All 's' should be zero. (mutter: Aha! I knew it) Solve b_sub_i for 1..n-2 (Grad student).
7) Substitute another power series to get an equation. (The grad students are gone, ask your hallmates, one of 'em has to be a math major.)
8) Let b_sub_n-1 equal the determinant of a funky, unexplained matrix (here, have an aspirin).
9) Everyone else in the class is out drinking by now, so don't worry about the next matrix, it's even funkier. Write a note on your hand to memorize it this weekend. Go drinking with peers.
10) Wake up at 3pm tomorrow, and try to remember what the hell all those squiggles meant.
11) Change your minor from math to polisci. Don't worry about taking Calc 1-3, DiffEq, or linear algebra. Note: many girls do not care about the roots of arbitrary polynomials, so no worries there. 8^)
"A witty saying proves nothing." ~Voltaire
"d'Oh!" ~Homer
Less Drang.