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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?

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Where is $p$ a continuous function from $\mathbb{R}$ and to $k[x_1,\ldots,x_n]$? In that case, what topology are you putting on $k[x_1,\ldots,x_n]$ - I don't think $k$ has a topology a priori, so why should $k[x_1,\ldots,x_n]$? –  Zev Chonoles Mar 13 '11 at 1:37
Sorry- say $k$ has a topology introduced by the norm $||x||$, which is the norm I use in the definition of the functional $L[f]$. And what I really meant was that $p$ was a homotopy between polynomials in $k[x_1,...,x_n]$. –  Alex Becker Mar 13 '11 at 3:56

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