2
$\begingroup$

(All polynomials here are assumed to be univariate)

I know that elementary solutions in terms of the coefficients may not exist for polynomials of order 5 or higher, but Wikipedia says that quintic and sextic equations could be done in analytic form using special functions such as the Jacobi theta.

My question is- do such analytic (not necessarily elementary) solutions of even higher polynomial equations also exist?

$\endgroup$
1
  • 2
    $\begingroup$ Camille Jordan apparently showed that any algebraic equation (one variable) can be solved using modular functions in his 1870 book Traité des Substitutions et des Équations Algébriques. I don't know anything about this, but some historical discussion of slightly prior work involving "explicit" solutions to the quintic, sextic, and septic equations can be found in Beyond the Quartic Equation by Robert Bruce King (1996). (The last chapter discusses sextic, septic, and Jordan's work.) $\endgroup$ Jul 24, 2019 at 10:09

1 Answer 1

5
$\begingroup$

More a long comment than an answer. The solution to the problem of finding an analytic expression for the roots of the general polynomial equation of order $n$ $$ a_n z^n + a_{n-1} z^{n-1} + \ldots + a_1 z + a_0=0 \label{1}\tag{1} $$ was, according to Giuseppe Belardinelli ([1] pp. 3-4), triggered by the discovery of hypergeometric functions of $n$ variables. Precisely, by using such classes of functions, two analytic formulas, one by Hjalmar Mellin and one by Giuseppe Belardinelli himself, were found in 1921: a brief description of their approaches is given below.
Mellin (see [2] and also [1], §20, where the transformation of \eqref{1} in \eqref{1a} is explicitly described) starts by considering an equivalent form of equation \eqref{1}, precisely the following one $$ z^n+b_1 z^{n_1}+\ldots+b_mz^{n_m} - 1=0,\label{1a}\tag{1a} $$ where $n_i<n$ and $b_j\neq 0$ for each $j=1,\ldots,m<n-1$. This form has the property that if one of its solution solutions is known, say $z_p=z_p(b_1,\ldots,b_m)$, it is possible to find all the remaining ones by the following formula $$ z_i(b_1,\ldots,b_m)=\zeta_iz_p(\zeta_i^{n_1}b_1,\ldots,\zeta_i^{n_p}b_m)\quad i=1,\ldots, n\label{2}\tag{2} $$ where $\zeta_i$, $i=1,\ldots, n$ are the solutions of the cyclotomic equation $$ \zeta^n-1=0 $$ Then Mellin succeeds in finding a particular hypergeometric function $z_p$, which he calls solution principale, such that $z_i(b_1,\ldots,b_m)$ is a root of \eqref{1a}, and thus by \eqref{2} he obtains a closed form analytic expression for the roots of \eqref{1}.
The solution of Belardinelli is similar in that it relies again on hypergeometric functions, but it is conceptually different: following the footsteps of Alfredo Capelli, he seeks the $n$ roots $z_i$, $i=1,\ldots, n$ of \eqref{1} in the form of convergent power series: and he succeeds in this task by defining a multiple Lagrange series of $n$-th order Pochammer hypergeometric functions that gives exactly the roots of \eqref{1} as a function of the coefficients $a_0,\ldots,a_n$ for each finite $n$ (see [1] for the details).

Additional note

The only English language reference I was able to find on the same topic is the short (translated from the Russian) paper [3] of E. N. Mikhalkin, which builds his result on the one of Mellin. He gives a very simple form for $z_p(b)$ (theorem 1, formula 4 at p. 302 of [3]): however, his formula is under all aspects a simplification of the one given by Mellin's in reference [2], and he doesn't explain nor describes how Mellin gets its result, nor provides any historical survey.

Bibliography

[1] Giuseppe Belardinelli (1960), Fonctions hypergéométriques de plusieurs variables et résolution analytique des équations algébriques générales (French), Mémorial des sciences mathématiques, no. 145 (1960), 80 p., MR121518, Zbl 0097.05901.

[2] Hjalmar Mellin (1921), Résolution de l’équation algébrique générale à l’aide de la fonction gamma (French), Comptes Rendus Hebdomadaires des Séances de l’Académie des Sciences, Paris, 172, pp. 658-661, JFM 48.1238.02.

[3] Evgenii Nikolaevich Mikhalkin, On solving general algebraic equations by integrals of elementary functions, Siberian Mathematical Journal 47, No. 2, 301-306 (2006). MR2227983, ZBL1115.33001.

$\endgroup$
4
  • $\begingroup$ Since I can't read French, I'll ask the question here: Do these constructions give insight into the difference between small $n$ and $n\geq 5$? The hope being that, when $n<5$, the nature of the solution is such that a solution by radicals is unsurprising (if still quite painful). $\endgroup$ Sep 6, 2019 at 18:00
  • $\begingroup$ @Semiclassical, both of the two Authors seem uninterested to show the equivalence of their general approach to the classical case $n\le 4$ therefore,in the cited references above, they do not show any explicit calculation pertaining to nor seem to discuss this point. However, even if the form of the solutions seems daunting, it may be that in the classical case they explicitly simplifiy soo much that their algebraic structure emerges evidently, though I am not so expert in the theory of hypergeometric functions of several variables to confirm or disprove this assertion, $\endgroup$ Sep 6, 2019 at 22:18
  • $\begingroup$ I can't read French (I can only read Korean and English, and simple Japanese texts w/ help from Google), I tried to read only the mathematical formulas but that was harder than I thought. $\endgroup$ Oct 25, 2019 at 13:52
  • $\begingroup$ @JustAYoungArtist I am not aware of sources written in English. However I will search the ZbMath if there’s some complete publication on the topic. $\endgroup$ Oct 25, 2019 at 14:28

You must log in to answer this question.

Not the answer you're looking for? Browse other questions tagged .