Is there a proof that there is no general method to solve transcendental equations?

Question

Being motivated by this post, I was wondering if there is a proof (analogous to the case of Diophantine equations) that there is no general method for solving transcendental equations? It seems pretty clear, intuitively, that there can be no general method; but the only reason I feel strongly about that is because I can't conceive that transcendental equations have a general method, but Diophantine equations don't. I was never able to understand the proof for the case of Diophantine equations, so I am not in a position to even know where to begin thinking about this. Has any work been done on this problem?

Answer 1

Yes. The problem is that the term "transcendental function" is absurdly general. For example, pick a computable bijection between the integers and the set of Turing machines and consider the function $f : \mathbb{R} \to \mathbb{R}$ which is $0$ on the integers corresponding to halting Turing machines, $1$ on the integers corresponding to non-halting Turing machines, and smoothly interpolated in between (for example via bump functions) such that if $x$ is not an integer then $0 < f(x) < 1$. Then the problem of determining the solutions to $f(x) = 1$ is equivalent to the halting problem. (And $f$ is even smooth.)

So to get any reasonable answer to this question one must pick a much more specific definition of transcendental function. One must also specify what it means to find a solution to an equation; is it enough to give an algorithm which will compute it to arbitrary accuracy, or must we get the answer in "closed form" (whatever that means)? There are a lot of issues here.

Answer 2

Intuition may be misleading here. In fact transcendental cases are often much easier than the integer Diophantine case. For example below is a table listing the known decidability results in various rings for Hilbert's tenth problem and the full first order theory, excerpted from Bjorn Poonen's interesting paper Hilbert's tenth problem over rings of number-theoretic interest