closed-form expression for roots of a polynomial

Question

It is often said colloquially that the roots of a general polynomial of degree $5$ or higher have "no closed-form formula," but the Abel-Ruffini theorem only proves nonexistence of algebraic closed-form formulas. And I remember reading somewhere that the roots of quintic equations can be expressed in terms of the hypergeometric function.

What is known, beyond Abel-Ruffini, about closed-form formulas for roots of polynomials? Does there exist a formula if we allow the use of additional special functions?

Transcendent functions do the job sometimes. For example, the roots of $X^n - 1$ are given by $e^{2 \pi k / n}$. — Hans Giebenrath, Feb 01 '13 at 06:50
@HansGiebenrath, roots of $X^n -1$ count as obtained by extraction of radicals. — Andreas Caranti, Feb 01 '13 at 07:22
Indeed the Galois group of $X^n-1$ is not only solvable, but also abelian. — JSchlather, Feb 01 '13 at 07:26
I'm sorry. You are right. I was aiming at CM-fields and extensions thereof generated by values of $j$. I think it should be possible to construct a non-solvable extension generated by the value of $j$. — Hans Giebenrath, Feb 01 '13 at 09:32
Many special cases of polynomial functions are solvable with inverse beta regularized $\text I^{-1}_s(a,b)$ if $a,b\in\Bbb N$. Would you like this answer? — Тyma Gaidash, Sep 02 '22 at 00:47

score 19 · Accepted Answer · answered Apr 06 '13 at 17:51

As you might already know, solutions to the quintic can be expressed in terms of either ${}_4 F_3$ hypergeometric functions or Jacobi theta functions. See King or Prasolov/Solovyev for details.

For polynomials of higher degree, there is also a general formula for the roots, due to Umemura. The formulae involve the multidimensional generalization of the Jacobi theta functions (the Riemann theta function), and are a bit unwieldy; see Umemura's paper if you want more details. See also this preprint for a solution of the reduced polynomial equation $x^n-x-\alpha=0$ in terms of hypergeometric functions.

score 8 · Answer 2 · edited Aug 20 '15 at 20:23

The question is based on

general polynomial of degree 5 or higher have "no closed-form formula"

This is not exactly true, what it should be said is that "general algebraic equations of degree higher than 4 do not admit solutions by radicals" which means that they cannot be solved by operations implying combinations of ordinary additions, multiplications, divisions, raising to powers, root extractions... On the other side it was shown by Hermite that fifth degree equations can be solved using the modular elliptic functions, which provide a generalization of the so called trigonometric solution of eqs. with degree lower than 5. Higher order (than 5) algebraic equations can be solved by employing other forms of elliptic functions.

In more recent times the use of the Lagrange inversion formula has allowed solutions in terms of hypergeometric functions. This technique was developed by an italian mathematician G. Belardinelli in 1959 and later rediscovered by M. L. Glasser in 2000 J. Comp. Appl. Math. 118 (2000) 169-171.

score 2 · Answer 3 · answered Sep 11 '21 at 09:14

You are right: "no closed-form formula" is not the right term here, because "closed form" means allowed constants, functions and operations from a given set.

Here are general closed-form solution formulas for quintics:
https://en.wikipedia.org/wiki/Bring_radical#Solution_of_the_general_quintic
How to solve fifth-degree equations by elliptic functions?

I have an addition to the other answers and the generally known:
The explicit elementary numbers are generated from the rational numbers by applying finite numbers of $\exp$, $\ln$ and/or radicals. Clearly, the elementary numbers contain all algebraic numbers and the explicit elementary numbers contain all explicit algebraic numbers (the numbers representable by radicals).
Chow [Chow 1999] gives his Corollary 1:
"If Schanuel's conjecture is true, then the algebraic numbers in" the explicit elementary numbers "are precisely the roots of polynomial equations with integer coefficients that are solvable in radicals."
That means, the algebraic equations that cannot be solved by radicals cannot be solved by elementary numbers (means by applying elementary functions).
$\ $

[Chow 1999] Chow, T.: What is a closed-form number. Am. Math. Monthly 106 (1999) (5) 440-448

score 1 · Answer 4 · answered Apr 10 '21 at 19:19

Polynomial equations can be solved with the 4 arithmetic operations, prime power roots and a suitable collection of other special roots obtained from a subset of the "unsolvable" equations of 5 and higher order. For the quintic equation, if I recall correctly, one needs only add the "star-root" x = *√c to x⁵ + x = c of real numbers c to the list of operations.

That's described in passing in R. Bruce King, Beyond the Quartic Equation, Birkhäuser, Boston, Basel, Berlin, 1996.

closed-form expression for roots of a polynomial

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