Let $p(t,x_1,...,x_n)$ be a continuous function which, evaluated at a real number $t$, gives a polynomial in $n+1$ variables over an algebraically closed field $k$. Let $V(t)$ be the vanishing set in affine $n$ -space of the ideal $(p(t,x_1,...,x_n))$. Is there any way to determine a map from $V(t_1)$ to $V_(t_2)$ by looking at $p(t,x_1,...,x_n)$? Furthermore, if we look at $f: V(t) \rightarrow V(t + dt)$, and define a functional $L[f]$ as the integral of $||f(x_1,...,x_n) - (x_1,...,x_n)||$ over a finite region $D$ , can we find the minimal such $f$ at each $t$ and from these construct the function $g: V(t_1) \rightarrow V(t_2)$?
Also, if the above are possible for an algebraically closed field, what would happen if this field where replaced with a non-commutative division ring (which is still algebraically closed), such as the quaternions?