Which kinds of zeroing equations of elementary functions can have solutions which are elementary numbers or explicit elementary numbers?
Take the equation
$$F(x)=0,$$
wherein $F$ is an elementary function.
Looking if F is elementary invertible is not the whole answer.
The Elementary functions are the functions which are generated from one variable by finite nesting of algebraic functions, $\exp$ and/or $\ln$.
The field of the Elementary numbers (Liouville numbers) is the smallest subfield of $\mathbb{C}$ generated by $\mathbb{Q}$ that is algebraically closed under $\exp$ and $\ln$. The field of the Explicit elementary numbers is the smallest subfield of $\mathbb{C}$ generated by $\mathbb{Q}$ that is closed under $\exp$ and $\ln$. They are defined i.a. in [Lin 1983] and [Chow 1999] respectively.
[Lin 1983] and [Chow 1999] give an answer for exponential polynomials. They prove that the irreducible polynomials $P\in\mathbb{\overline{Q}}[x,e^x]$ involving both $x$ and $e^x$ cannot have solutions that are elementary numbers and explicit elementary numbers respectively.
My questions:
Are there answers for further classes of functions?
Can the methods of Lin and Chow be used for further classes of functions?
There is at least a hint. Chow writes: "The reader may check that our arguments generalize readily to other transcendental equations such as $x=cos\ x$."
If his method can be applied to the equation $x=cos\ x$, could it perhaps be applied also for the functions $R\in\overline{\mathbb{Q}}(x,t_1,...,t_n)$, wherein $1<n\leq 14$, $t_1=\exp(x)$, $t_2=\exp(xi)$, $t_3=\sin(x)$, $t_4=\cos(x)$, $t_5=\tan(x)$, $t_6=\cot(x)$ $t_7=\sec(x)$, $t_8=\csc(x)$, $t_9=\sinh(x)$, $t_{10}=\cosh(x)$, $t_{11}=\tanh(x)$, $t_{12}=\coth(x)$, $t_{13}=\text{sech}(x)$, $t_{14}=\text{csh}(x)$?
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[Chow 1999] Chow, T.: What is a closed-form number. Am. Math. Monthly 106 (1999) (5) 440-448
