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In the classic textbook Introduction to the Theory of Equations (Conkwright, 1941), on p. 85, the author writes that “the algebraic solution of the general equation of degree n is impossible if n is greater than four. By this we mean that it is not possible to find the exact values of the roots of every equation of degree n (n > 4) by performing upon the coefficients a finite number of additions, subtractions, multiplications, divisions, and root extractions.”

That proposition is considered proven by the Abel-Ruffini theorem of 1824, as Conkwright surely knew. Tantalizingly, though, he goes on to say – without elaborating – that “the general equation of degree n has been solved in terms of Fuchsian functions.” And there, it seems, the trail ends. A web search of about an hour has yielded nothing more than various restatements of the problem.

Two questions:

(1) Can anyone state, in a form suitable for reduction to a computer algorithm, a solution or family of solutions of the general equation of degree n (whether based on Fuchsian functions or not)?

(2) Would your solution(s) yield theoretically exact values, or only converging approximations?

This is my first post to your site, and I apologize in advance if I've overlooked answers right under my nose. These questions have, however, stumped my math department chair. (I’m on loan to him – I normally teach humanities, but I have an engineering background and I was asked to fill in for some math courses.)

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Spitballing here, but aren't the "Fuchsian functions" the same as the solutions to the Fuchsian second-order ODE as presented in dlmf.nist.gov/31.14 ? I am at least aware that these solutions to this ODE can degenerate to (generalized) hypergeometric functions, which also have been used in the solution of the general nth-degree equation. The alternative to these generalized hypergeometrics of course are the multivariate theta functions, as Charles Siegel has pointed out. –  J. M. Aug 10 '10 at 1:35
I think "Fuchsian functions" is a reference to modular functions and modular forms (possibly meromorphic). Hermite's famous solution for the quintic was described by Felix Klein in "Lectures on the icosahedron" and can be generalized to the sextic, but not to general $n$th degree equation (this is explained in the addenda to the Russian edition, including Serre's letter to Gray). Umemura's solution in theta functions mentioned by Charles works for general $n.$ –  Victor Protsak Aug 10 '10 at 2:04
Earlier references on MO: mathoverflow.net/questions/9474/…, mathoverflow.net/questions/32099/… Classical theory is carefully explained in Fricke's Algebra, II (in German). –  Victor Protsak Aug 10 '10 at 2:25
Wolfram has a nice poster at library.wolfram.com/examples/quintic about the quintic case –  Richard Borcherds Aug 10 '10 at 3:59
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3 Answers

Fuchsian functions are described on wikipedia

(1) Can anyone state, in a form suitable for reduction to a computer algorithm, a solution or family of solutions of the general equation of degree n (whether based on Fuchsian functions or not)?

I had prepared notes on this topic sometime ago so here goes! There are three principal ways to solve an algebraic equation of degree n:

  1. Algebraically by using radicals in a number field. This method doesn't work if the Galois group of the equation is not solvable, which happens for general equations beyond degree 4. In terms of differential equations, the method is analogous to integrating by quadrature if the Automorphism group(Lie group) is solvable. For more on this, see Cox's book on Galois theory and Gaal's book on Galois theory Sec 4.5
  2. Transcendental - by reducing equation to some familiar "modular equation" : The genus of the function which is used as the solution generally depends on the degree of the equation. (See Hilbert's 13th problem) For degree {2, 3, 4}, you use trigonometric functions. For degree 5, (use genus 1) elliptic functions. For degree 6, you need genus 2 theta functions. For degree 7, depending on the Galois group, requires hyperelliptic or theta functions of genus 3. Beyond degree 7, you need to see the Umemura paper mentioned by Charles Seigel in his answer. I have omitted some information here but for general computational references, see Bruce King's book Beyond Quartic. and McKean Moll's book Elliptic Curves Chapter 5
  3. Solution by complex dynamics : You have to find an iterative map working in a function field such that attractor(map)=root(equation). The Newton-Raphson method which we apply for degree 2 and 3 equations is actually a primitive version of this method. You can see the Julia set for Newton's method on wolfram. For more of this method, see the papers of Scott Crass and Doyle McMullen. A general reference for the method of complex dynamics is Shurman's book Geometry of the quintic. He illustrates the similarities between this method and the method of adjoining radicals in a number field. In both cases, you are building an extension over a number/function field. The number field criterion says that says that splitting field can be constructed if Galois group is solvable while the function field criterion says the that Galois group must be nearly solvable (i.e. nearly solvable implies factor groups must be Mobius groups - rotation groups of the sphere)
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I just discovered a previous related question mathoverflow.net/questions/23094/… –  SandeepJ Aug 10 '10 at 2:26
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I'm not familiar with Fuchsian functions, but at the end of Mumford's "Tata Lectures on Theta II" there's an article by Hiroshi Umemura which explains how to arrive at exact solutions using theta functions. Part of it is available on Google Books here.

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you can see this paper:on fuchsian difffrential equations reducible to hypergeometric equationsby linear transformations.By Tosihusa kimura. http://fe.math.kobe-u.ac.jp/FE/Free/vol13/fe13-17.pdf

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