I became interested in the following problems by studying kinematic configuration spaces:
Let
$C=V(f_1,\ldots,f_r)$ where $f_i\in \mathbb{R}[\vec{x},\vec{y}]$,
$J=V(g_1,\ldots,g_s)$ where $g_i\in \mathbb{R}[\vec{x}]$,
$p:C\to J$ be a surjective open polynomial map.
Find $h_1,\ldots, h_k\in \mathbb{R}[\vec x]$ such that
$$ S:= \{\vec x\in J : \dim p^{-1}(\vec x) >0\} = V(h_1,\ldots, h_k) $$
By the upper-semicontinuity of fiber dimension one can show that the set of points $\vec c=(\vec x,\vec y)\in C$ such that $\dim p^{-1}(p(\vec c)) = 0$ is Zariski open. Since $p$ is an open surjection it follows that the set of $\vec x\in J$ with finite fibers is Zariski open in $J$. Thus the locus of points in $J$ with positive dimensional fibers must be a subvariety of $J$. This justifies the existence of $h_1,...,h_k$.
Is there an algorithmic way to determine a generating set for $S$ that could be implemented in a computer algebra system?
I imagine this will involve some combination of things like Jacobians, rank, determinants and Gröbner basis. If it helps, in my case the map $p$ is the projection $p(\vec x,\vec y)=\vec x$ restricted to $C$.
Similarly, consider $$ S_d:= \{\vec x\in J : \dim p^{-1}(\vec x) \geq d\}. $$ Are there practical algorithmic methods to determine a generating set for $S_d$?