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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."
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.
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.
Now I'll RTFA to determine which it really is....
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.)
Using series to approximate the solution of differntial equations is taught in class. Heck, go a little further in mathematics and you'll conjure up polynomials functions as the solution to a set of partial differential equations, known as the Galerkin Method
So in what way is the above news? (Hint, take a look at the link and what's stated there.)
The cesspool just got a check and balance.
It's a shame the paper doesn't say why this is better than using Sturm sequences, which work perfectly well.
What I'm surprised at, though, is that nobody's pointed out the most obvious problems with this scheme:
- Your polynomial has (up to) n roots; this approach converges, maybe, to one of them. Which one do you get, and how do you get the others?
- For that matter, some chunk of those n roots may be complex; but all the maths in his article are real. How do you solve, say, x^6+1 = 0?
The change in radius of convergence at the end of his post is a little dicey too, at least as I'm reading it, but I could be misreading. Still, I'm frankly stunned. Worthy of a press release? If I'd turned this in as a school assignment it wouldn't even have been worth an A!