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Suppose I have a specific equation, not a general form like "all polynomials" and I want to determine whether it's solvable in terms of algebraic operations and logarithms, is there a method to carry this out on a case by case basis?

Maybe even simpler if not the above, is there any way to use differential algebra on just the solvability of $y=f(x)$ if $f(x)$ is elementary?

RandomWordMashup
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  • What is your definition of elementary? A useful tool is the implicit function theorem https://en.wikipedia.org/wiki/Implicit_function_theorem however this does not tell you what your solution looks like. Also I think you mean $f(x) = 0$ (or another constant for that matter) since $y = f(x)$ refers to the graph of $f$. But I do think the question is a little too vague for a good answer. – Dayton Mar 02 '20 at 21:25
  • I'm using the conventional definition of elementary being comprised of elementary functions. I guess I won't rule out the implicit function theorem but I am not seeing exactly how it relates to the question seeing f(x) is a function of one variable. I put galois theory as a tag, which is meant to give some insight that it is comparable to the solvability of polynomials. – RandomWordMashup Mar 02 '20 at 21:32
  • Okay, the implicit function theorem may not apply here then for one variable. I still think you may be out of luck though unless you are more specific. For example $f(x) = e^{x} = 0$ will have no solution over any field, unlike polynomials that will always have solutions in $\mathbb{C}$. – Dayton Mar 02 '20 at 21:44
  • Okay, but what if I'm not solving $e^x = 0$, what if I'm solving $e^x = 1$? Is there a kind of field argument to be made for that? – RandomWordMashup Mar 02 '20 at 21:45
  • Maybe this post will help https://math.stackexchange.com/questions/982984/prove-that-an-equation-has-no-elementary-solution – Dayton Mar 02 '20 at 21:49
  • I guess that sort of helps, it provides more insight, but the post doesn't really answer anything. Transform how? What even is a partial inverse? – RandomWordMashup Mar 02 '20 at 21:52

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