A set of numbers $S=\{x_1,...,x_n\}$ is said to be algebraically dependent if there exists a (multivariate) polynomial $p$ with coefficients in $\mathbb Q$ whose roots contain $x_1,...,x_n$ (or a subset thereof).

Given numbers $S=\{x_1,...,x_n\}$, is it decidable whether they are algebraically dependent?

Now, there is obviously an encoding issue here, since the numbers need to be transcendental, so describing them is problematic. Any solution would do, but specifically, I am interested in the case where all the numbers are of the form $e^{a_i}$, where the $a_i$'s are linearly independent over $\mathbb Q$ (but could be algebraic).

The motivation for this question is understanding whether a (positive) solution to Schanuel's conjecture would automatically lead to a constructive solution.

A set of numbers $S=\{x_1,...,x_n\}$ is said to be algebraically dependent if there exists a (multivariate) polynomial $p$ with coefficients in $\mathbb Q$ whose roots contain $x_1,...,x_n$ (or a subset thereof).

Given numbers $S=\{x_1,...,x_n\}$, is it decidable whether they are algebraically dependent?

Now, there is obviously an encoding issue here, since the numbers need to be transcendental, so describing them is problematic. Any solution would do, but specifically, I am interested in the case where all the numbers are of the form $e^{a_i}$, where the $a_i$'s are linearly independent over $\mathbb Q$.

The motivation for this question is understanding whether a (positive) solution to Schanuel's conjecture would automatically lead to a constructive solution.

UPDATE (based on the comments): the encoding of the numbers is an issue. I'll accept any answer which suggests either a reasonable encoding for a subset of the reals for which the problem is non-trivial, or a solution that uses a computational model that allows real numbers.

A set of numbers $S=\{x_1,...,x_n\}$ is said to be algebraically dependent if there exists a (multivariate) polynomial $p$ with coefficients in $\mathbb Q$ whose roots contain $x_1,...,x_n$ (or a subset thereof).

Given numbers $S=\{x_1,...,x_n\}$, is it decidable whether they are algebraically dependent?

Now, there is obviously an encoding issue here, since the numbers need to be transcendental, so describing them is problematic. Any solution would do, but specifically, I am interested in the case where all the numbers are of the form $e^{a_i}$, where $a_i$ is rational (so we can describe the numbers by $a_i$).

The motivation for this question is understanding whether a (positive) solution to Schanuel's conjecture would automatically lead to a constructive solution.

A set of numbers $S=\{x_1,...,x_n\}$ is said to be algebraically dependent if there exists a (multivariate) polynomial $p$ with coefficients in $\mathbb Q$ whose roots contain $x_1,...,x_n$ (or a subset thereof).

Given numbers $S=\{x_1,...,x_n\}$, is it decidable whether they are algebraically dependent?

Now, there is obviously an encoding issue here, since the numbers need to be transcendental, so describing them is problematic. Any solution would do, but specifically, I am interested in the case where all the numbers are of the form $e^{a_i}$, where the $a_i$'s are linearly independent over $\mathbb Q$ (but could be algebraic).

The motivation for this question is understanding whether a (positive) solution to Schanuel's conjecture would automatically lead to a constructive solution.