Abel-Ruffini says that there is no closed form solution for $5$'th degree polynomials. I am interested in which cases something like this occurs. Suppose we have a bunch of polynomial equations with parameters, when can we find the solutions for given parameters? So the quintent situation has one polynomial equation, 6 parameters and 1 unknown
$$a_6 x^5 + a_5x^4 + a_4 x^3 + a_3 x^2 + a_2 x + a_1 = 0$$
Attempt to generalise
Suppose we have a bunch of polynomials $f_1, ..., f_n: \mathbb{Q}^m \to \mathbb{Q}$ with $k$ parameters. Is there a computable function that assigns to $(a_1, ..., a_k)$ the Gödel number of some computable function which enumerates all $(x_1, ..., x_{m - k})$ such that $(a_1, ..., a_k, x_1, ..., x_{m - k})$ is a zero for all $f_i$'s?
Actual question
Are there techniques to answer this question negatively if you have explicit examples of $f_1, ..., f_n$, for example by reducing it to a problem which is known to be unsolvable? For high enough degree polynomials like
$$a_1 x_1^5 - a_1 x_1 - a_2 = 0 $$
$$x_6^2 + x_2 = 0 $$
this is easy by reducing it to a fifth degree polynomial equation (take $a_1 = a_2 = 1$, then it gives you a closed form for the solutions of $x^5 - x - 1 = 0$), but what if the degree is low like
$$xy = 1$$
$$xzk + 5 - 8dj = 0$$
$$abc + xy - 8d = 0 $$
?