# Closed-form solvability of elementary transcendental equations?

Fern-Ching Lin ([Lin 1983]) and Timothy Chow ([Chow 1999]) asked, when the solutions of a transcendental equation of elementary functions can be elementary numbers.

My question is:
To which more general kinds of transcendental equations can Lin's theorem be extended or generalized?

$$\mathbb{L}$$ denotes the Liouvillian numbers (= Elementary numbers). The Elementary numbers are subdivded into the Explicit elementary numbers $$\mathbb{E}$$ and the Implicit elementary numbers.

Lin's theorem:
If Schanuel's conjecture is true and $$P(X,Y)\in\overline{\mathbb{Q}}[X,Y]$$ is an irreducible polynomial involving both $$X$$ and $$Y$$ and $$P(z_0,e^{z_0})=0$$ for some nonzero $$z_0\in\mathbb{C}$$, then $$z_0$$ is not in $$\mathbb{L}$$.

A corollary of Lin's theorem is the conclusion "then $$z_0$$ is not in $$\mathbb{L}$$ and not in $$\mathbb{E}$$", because $$\mathbb{E}\subset\mathbb{L}$$.

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[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

Theorem 1.
Let $$E$$ be an elementary function with $$E\colon \mathrm{dom}(E)\subseteq\mathbb{C}\to\mathbb{C}$$, $$z\in \mathrm{dom}(E)$$ and $$E(z)\neq 0$$.
If Schanuel's conjecture is true and $$P(X,Y)\in\overline{\mathbb{Q}}[X,Y]$$ is an irreducible polynomial involving both $$X$$ and $$Y$$ and $$P(E(z),e^{E(z)})=0$$ for some $$z\in\mathbb{C}$$, then $$z$$ is not in $$\mathbb{L}$$ and not in $$\mathbb{E}$$.

Proof.
Lin's theorem and the corollary mentioned in the question imply $$E(z)$$ is not in $$\mathbb{L}$$ and not in $$\mathbb{E}$$.
Because $$E$$ is an elementary function, $$E(z)\in\mathbb{L}$$ for all $$z\colon z\in \mathrm{dom}(E) \land z\in\mathbb{L}$$.
But because $$E(z)$$ is not in $$\mathbb{L}$$ and not in $$\mathbb{E}$$ as stated above, $$z$$ is not in $$\mathbb{L}$$ and not in $$\mathbb{E}$$. $$\ \ \ \square$$
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Let
$$E,E_1$$ be non-constant elementary functions,
$$P\in\overline{\mathbb{Q}}[x,y]\setminus(\overline{\mathbb{Q}}[x]\cup\overline{\mathbb{Q}}[y])$$ irreducible over $$\mathbb{C}$$,
$$p\in\overline{\mathbb{Q}}[x]\setminus\overline{\mathbb{Q}}$$,
$$R\in\overline{\mathbb{Q}}(x,y)\setminus(\overline{\mathbb{Q}}[x]\cup\overline{\mathbb{Q}}[y])$$ in reduced form, with numerator irreducible over $$\mathbb{C}$$,
$$r\in\overline{\mathbb{Q}}(x)\setminus\overline{\mathbb{Q}}$$.

$$P(E(z),e^{E(z)})=0,\ \ \ \ P(E(z),\ln(E(z)))=0$$ $$R(E(z),e^{E(z)})=0,\ \ \ \ R(E(z),\ln(E(z)))=0$$ $$E(R(E_1(z),e^{E_1(z)}))=E(0),\ \ \ \ E(R(E_1(z),\ln(E_1(z))))=E(0)$$