Consider a system of polynomial equations $S$ in multiple variables $x_1,\dots,x_n$ over the field $\mathbb{C}$.
Is there a simple characterization of when the following property holds:
There exists a set of polynomials $f_1,\dots,f_n$ in variables $y_1,\dots,y_m$ such that:
1) For all assignments of $y_1,\dots,y_m$, the assignment $x_1=f_1,\dots,x_n=f_n$ satisfies $S$
2) For all satisfying assignments of $S$ there exists an assignment to $y_1,\dots,y_m$ such that $f_1=x_1,\dots,f_n=x_n$
I'd particularly like to know if this property has a name or has been studied at all, and especially if there's a method to find such a set if one exists!
A very simple example of this is to consider the determinant of a rank one 2 by 2 matrix: $m_{1,1}\cdot m_{2,2}-m_{1,2}\cdot m_{2,1}=0$. By considering the SVD of such a matrix we can get the following:
$m_{1,1}=y_1\cdot y_3$
$m_{1,2}=y_2\cdot y_3$
$m_{2,1}=y_1\cdot y_4$
$m_{2,2}=y_2\cdot y_4$
for which the property above clearly holds.
I have very little formal instruction in algebraic geometry or related fields, so it's possible I'm lacking some basic concepts to recognize this! Apologies if the title doesn't clearly represent what I'm asking.