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When we deal with algebraic equations involving polynomial and so on we know what the degree of the equation is and this tells us how many solutions we'll find (at least in complex numbers). But this is not all, the degree gives us important information if it is solvable or not by radical (a quartic equation could be complicated but it is always possible while a six degree equation that seems easier could be not).

On the other hand when we work with transcendental equation we also have an infinity of solution (most of the times in complex numbers) but my question is: can we talk about the degree of a transcendental equation ?

An equation like $xe^x=k$ is really trivial to solve using Lambert $W$-function while an equation like $(x^2+2x+3)e^x=k$ is more hard to solve so is there any general method that tells us something like "this can be solved with this special function" or "this equation has exactly 4 real root even if we don't know how to find them" ? Or maybe every equation has to be studied by itself... ?

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We speak about solving of equations in closed form. "Closed form" means expressions of allowed functions (Wikipedia: Closed-form expression). If an equation is solvable in closed form depends therefore on the functions you allow.

A general method for solving a given equation $H(x)=0$ is to apply the compositional inverse $H^{-1}$ of $H$: $x=H^{-1}(0)$. In general, $H$ and $H^{-1}$ are correspondences. But often it is possible to split the problem into subproblems where $H$ and $H^{-1}$ are functions. For applying this method, $H$ and $H^{-1}$ have to be known. That means they have to be in closed form.

1.)

Because the identity function is an algebraic function, each transcendental function $H$ as given above can be expressed as a composition of an outer univariate or multivariate algebraic function and one or more inner transcendental functions $f_{1}$, $f_{2}$, ..., $f_{n}$. You can determine the degree of the outer algebraic function.

2.)

For the elementary functions, there is a structure theorem of J. F. Ritt (mentioned below) that can help to decide if a given kind of equations of elementary functions can be solved by transforming a given equation by applying only elementary functions.

The elementary functions are according to Liouville and Ritt those functions of one variable which are obtained in a finite number of steps by performing algebraic operations and taking exponentials and logarithms (Wikipedia: Elementary function).

The incomprehensibly unfortunately hardly noticed theorem of Joseph Fels Ritt in [Ritt 1925] Ritt, J. F.: Elementary functions and their inverses. Trans. Amer. Math. Soc. 27 (1925) (1) 68-90 answers which kinds of Elementary functions can have an inverse which is an Elementary function.

The problem of solving a given ordinary equation in a differential field is solved in Rosenlicht, M.: On the explicit solvability of certain transcendental equations. Publications mathématiques de l'IHÉS 36 (1969) 15-22.

If $H$ can be decomposed into compositions of algebraic functions and other known Standard functions than $\exp$ and $\ln$, an analog theorem to the theorem of Ritt of [Ritt 1925] could be applied. I hope to prove such a generalization of Ritt's theorem for this class of functions.

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  • $\begingroup$ +1 Very informative. I wonder if it answers the OP's soft question. And good luck with your quest for a proof. $\endgroup$ – Ethan Bolker Oct 28 '17 at 17:28
  • $\begingroup$ The answer gives some hints for methods for deciding which kinds of equations can have a closed-form solution and which not. It answers some parts of the OP's question. $\endgroup$ – IV_ Oct 28 '17 at 17:39

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