This is the a-part of the exercise 10 (pp. 141) in Ideals, Varieties and Algorithms.
Let $k = \mathbb{C}$ and define a curve $\gamma(t) = (f_1(t),\dots,f_n(t)$, $f_1,\dots,f_n\in k[t]$. I want to show that the parametrized equations fill up all the variety $\mathbb{V}(I_1) = \mathbb{V}\left(\left<x_1 - f_1(t),\dots,x_n - f_n(t)\right>\bigcap k[x_1,\dots,x_n]\right)$, where $I_1$ is the first elimination ideal of $I$. My problem is that I am not quite sure how to start the proof:
We know from the Polynomial Implicitization Theorem that if $F:k^m\to k^n, (t_1,\dots,t_m)\mapsto (f_1(t_1,\dots,t_m),\dots,f_n(t_1,\dots,t_m))$, then $F(k^m) = \pi_m\left(\mathbb{V}\left(x_1 - f_1,\dots,x_n - f_n\right)\right)$ where $\pi_m$ is the projection onto the last $(n + m) - m = n$ components, and $\mathbb{V}(I_m)$ is the smallest variety in $k^n$ containing $F(k^m)$. Therefore, I think that what I would need to show is that $F(k^1) = \mathbb{V}(I_1)$, but currently, the only thing I know about $I_1$ is that its elements are free from $t$. This also happens to be the point where I am stuck: I know that $I_1$'s elements are free from $t$, but I don't know how to relate it to the parametric equations. So how should I proceed with the proof?
You have to use the Extension theorem. In my copy of the book, it is Corollary 4 in Chapter Elimination Theory, subsection "The Elimination and Extension Theorems". I state it here:
Let $I = \langle f_1,\dots, f_n\rangle\subset \mathbb{C}[x_1,\dots,x_n]$, and assume that for some $i$, $f_i$ is of the form $$ c x_i^N + \text{ terms in which } x_i \text{ has degree } < N $$ where $c \in \mathbb{C}$ is nonzero and $N > 0$. If $I_1$ is the first elimination ideal, and $(a_2,\dots,a_n) \in V(I_1)$, then there exists $a_1 \in \mathbb{C}$ such that $(a_1,\dots,a_n) \in V(I)$.
Just notice that either all polynomials $f_1,...,f_n$ are constant, so the image of the parametrization is just one point, or at least one is non-constant. Then you can use the Extension Theorem.
You can also look at the section "Implicitization", at the paragraph after the proof of "Polynomial implicitization" theorem, where the authors use the Extension theorem to show that the surface given by $$ x = t + u, y = t^2 + 2tu, z = t^3 + 3t^2u $$ fills up all of $V(x^3z - (3/4)x^2y^2 - (3/2)xyz + y^3 + (1/4)z^2)$.