Reverse search for rational function

Question

Say we have two transcendental numbers, u and v. And u presumably can be obtained as a result of applying a rational function $Q$ with integer coefficients to v. Is it possible to find such rational function?

In other words we need to find two polynomials $P_1$ and $P_2$ with integer coefficients such that

$u=\frac{P_1(v)}{P_2(v)}$

Answer 1

Let's ask a much simpler question: if we have two transcendental numbers, and their difference is a rational number, can we find that rational number? Seems to me it would depend a bit on what it means to "have" a transcendental number. For all we know, $\pi-e$ is rational. The more decimals we know in the expansions of $\pi$ and $e$, the better the lower bound we can put on the numerator and denominator of the rational, but how can we ever find the rational?

Answer 2

Let's write it as $P_1(v)u = P_2(v)$, or rather $$ \sum \alpha_i v^i u - \sum \beta_i v^i = 0. $$ Now use an integer relation algorithm.