# Why are there elementary equations that are not solvable in closed form?

Elementary equations and closed-form solutions can be important in mathematics and in the natural sciences, e.g. for discovering and describing certain relationships.

$$\log\colon x\mapsto\log(x)$$; $$x\neq 0$$; $$-\pi<\Im(\log(x))\leq\pi$$ for all $$x$$

$$\mathbb{L}$$ denote the Liouvillian numbers (= Elementary numbers). $$\mathbb{L}$$ is the smallest field that contains $$\mathbb{Q}$$ and is closed under algebraic operations, $$\exp$$ and $$\log$$. The Elementary numbers are divided in the explicit elementary numbers and the implicit elementary numbers.
$$i\in\mathbb{L}$$

In [Chow 1999], Chow cites [Lin 1983] and writes:
"Let $$\overline{\mathbb{Q}}$$ denote the algebraic closure of $$\mathbb{Q}$$. Then Lin's result is the following.
Theorem 1. If Schanuel's conjecture is true and $$f(x,y)\in\overline{\mathbb{Q}}[x,\exp(x)]$$ is an irreducible polynomial involving both $$x$$ and $$y$$ and $$f(\alpha,\exp(\alpha))=0$$ for some nonzero $$\alpha\in\mathbb{C}$$, then $$\alpha\notin\mathbb{L}$$."

Chow continues: "The reader may check that our arguments generalize readily to other transcendental equations such as $$x=\cos\ x$$."


1. Can Lin's theorem be generalized to all irreducible polynomials $$f$$ with $$f(x,y,z)\in\overline{\mathbb{Q}}[x,\exp(x),\exp(ix)]$$ involving $$x$$, $$y$$ and $$z$$, or $$x$$ and ($$y$$ or $$z$$)?


2. Can Lin's theorem be generalized to all irreducible polynomials $$f$$ with $$f(x,y,z)\in\overline{\mathbb{Q}}[x,\exp(A_1(x)),\exp(A_2(ix))]$$ involving $$x$$, $$y$$ and $$z$$, or $$x$$ and ($$y$$ or $$z$$), wherein $$A_1$$ and $$A_2$$ are arbitrary algebraic functions?


3. To which kinds of Elementary equations can Lin's theorem further be generalized?
This question is necessary, because Ritt's and Risch's elementary invertibility of elementary functions (How to extend Ritt's theorem on elementary invertible bijective elementary functions) is not the whole answer for solving equations elementary.
$$\$$

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

[Lin 1983] Ferng-Ching Lin: Schanuel's Conjecture Implies Ritt's Conjectures. Chin. J. Math. 11 (1983) (1) 41-50

• What does "closed under log" mean? It is a subfield of $\mathbb{C}$ containing every determination of $\log(x)$ whenever it contains $x\neq 0$? – Gro-Tsen Jun 1 at 14:46
• I added this to my question now. Chow has therefore an extra definition for "closed under $exp$ and $log$". – IV_ Jun 1 at 14:55
• You added the clarification that the argument of $\log$ should be nonzero, but this was not the important part of my comment: my question is whether one demands that every determination of the logarithm (or equivalently, $2i\pi$) belongs to the field. – Gro-Tsen Jun 1 at 15:54
• @Gro-Tsen: That seems to follow from the OP's latest version, as $i\pi = \log (-1) - \log 1 \in L$ (or slightly modify this, depending on what is meant by "main branch"). – Christian Remling Jun 1 at 17:11
• Question 2 seems trivial as stated, since we can take $A_1(x)=ix,\ A_2(x)=x$. – Matt F. Jun 1 at 18:13